An Invitation to Random Schr¨odinger operators - FernUniversität in ...
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An Invitation to Random Schr¨odinger operators - FernUniversität in ...
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<strong>An</strong> <strong>Invitation</strong> <strong>to</strong><br />
<strong>Random</strong> Schröd<strong>in</strong>ger opera<strong>to</strong>rs<br />
Werner Kirsch<br />
Fakultät für Mathematik und Informatik<br />
FernUniversität Hagen<br />
D-58084 Hagen, Germany<br />
email: werner.kirsch@fernuni-hagen.de<br />
Abstract<br />
This review is an extended version of my m<strong>in</strong>i course at the États de la recherche:<br />
Opérateurs de Schröd<strong>in</strong>ger aléa<strong>to</strong>ires at the Université Paris 13 <strong>in</strong> June 2002, a<br />
summer school organized by Frédéric Klopp.<br />
These lecture notes try <strong>to</strong> give some of the basics of random Schröd<strong>in</strong>ger opera<strong>to</strong>rs.<br />
They are meant for nonspecialists and require only m<strong>in</strong>or previous knowledge<br />
about functional analysis and probability theory. Nevertheless this survey <strong>in</strong>cludes<br />
complete proofs of Lifshitz tails and <strong>An</strong>derson localization.<br />
Copyright by the author. Copy<strong>in</strong>g for academic purposes is permitted.<br />
1
3<br />
CONTENTS<br />
1. Preface 5<br />
2. Introduction: Why random Schröd<strong>in</strong>ger opera<strong>to</strong>rs 7<br />
2.1. The sett<strong>in</strong>g of quantum mechanics 7<br />
2.2. <strong>Random</strong> Potentials 7<br />
2.3. The one body approximation 10<br />
3. Setup: The <strong>An</strong>derson model 13<br />
3.1. Discrete Schröd<strong>in</strong>ger opera<strong>to</strong>rs 13<br />
3.2. Spectral calculus 15<br />
3.3. Some more functional analysis 18<br />
3.4. <strong>Random</strong> potentials 19<br />
4. Ergodicity properties 23<br />
4.1. Ergodic s<strong>to</strong>chastic processes 23<br />
4.2. Ergodic opera<strong>to</strong>rs 25<br />
5. The density of states 29<br />
5.1. Def<strong>in</strong>ition and existence 29<br />
5.2. Boundary conditions 35<br />
5.3. The geometric resolvent equation 40<br />
5.4. <strong>An</strong> alternative approach <strong>to</strong> the density of states 41<br />
5.5. The Wegner estimate 43<br />
6. Lifshitz tails 51<br />
6.1. Statement of the Result 51<br />
6.2. Upper bound 52<br />
6.3. Lower bound 56<br />
7. The spectrum and its physical <strong>in</strong>terpretation 61<br />
7.1. Generalized Eigenfunctions and the spectrum 61<br />
7.2. The measure theoretical decomposition of the spectrum 65<br />
7.3. Physical mean<strong>in</strong>g of the spectral decomposition 67<br />
8. <strong>An</strong>derson localization 75<br />
8.1. What physicists know 75<br />
8.2. What mathematicians prove 76<br />
8.3. Localization results 77<br />
8.4. Further Results 78<br />
9. The Green’s function and the spectrum 81<br />
9.1. Generalized eigenfunctions and the decay of the Green’s function 81<br />
9.2. From multiscale analysis <strong>to</strong> absence of a.c. spectrum 83<br />
9.3. The results of multiscale analysis 85<br />
9.4. <strong>An</strong> iteration procedure 86<br />
9.5. From multiscale analysis <strong>to</strong> pure po<strong>in</strong>t spectrum 89<br />
10. Multiscale analysis 93<br />
10.1. Strategy 93
4<br />
10.2. <strong>An</strong>alytic estimate - first try 95<br />
10.3. <strong>An</strong>alytic estimate - second try 100<br />
10.4. Probabilistic estimates - weak form 103<br />
10.5. Towards the strong form of the multiscale analyis 105<br />
10.6. Estimates - third try 106<br />
11. The <strong>in</strong>itial scale estimate 113<br />
11.1. Large disorder 113<br />
11.2. The Combes-Thomas estimate 113<br />
11.3. Energies near the bot<strong>to</strong>m of the spectrum 116<br />
12. Appendix: Lost <strong>in</strong> Multiscalization – A guide through the jungle 119<br />
References 121<br />
Index 127
5<br />
1. Preface<br />
In these lecture notes I try <strong>to</strong> give an <strong>in</strong>troduction <strong>to</strong> (some part of) the basic theory<br />
of random Schröd<strong>in</strong>ger opera<strong>to</strong>rs. I <strong>in</strong>tend <strong>to</strong> present the field <strong>in</strong> a rather self<br />
conta<strong>in</strong>ed and elementary way. It is my hope that the text will serve as an <strong>in</strong>troduction<br />
<strong>to</strong> random Schröd<strong>in</strong>ger opera<strong>to</strong>rs for students, graduate students and researchers<br />
who have not studied this <strong>to</strong>pic before. If some scholars who are already<br />
acqua<strong>in</strong>ted with random Schröd<strong>in</strong>ger opera<strong>to</strong>rs might f<strong>in</strong>d the text useful as well I<br />
will be even more satisfied.<br />
Only a basic knowledge <strong>in</strong> Hilbert space theory and some basics from probability<br />
theory are required <strong>to</strong> understand the text (see the Notes below). I have restricted<br />
the considerations <strong>in</strong> this text almost exclusively <strong>to</strong> the <strong>An</strong>derson model, i.e. <strong>to</strong> random<br />
opera<strong>to</strong>rs on the Hilbert space l 2 (Z d ). By do<strong>in</strong>g so I tried <strong>to</strong> avoid many of<br />
the technical difficulties that are necessary <strong>to</strong> deal with <strong>in</strong> the cont<strong>in</strong>uous case (i.e.<br />
on L 2 (R d )). Through such technical problems sometimes the ma<strong>in</strong> ideas become<br />
obscured and less transparent.<br />
The theory I present is still not exactly easy staff. Follow<strong>in</strong>g E<strong>in</strong>ste<strong>in</strong>’s advice, I<br />
tried <strong>to</strong> make th<strong>in</strong>gs as easy as possible, but not easier.<br />
The author has <strong>to</strong> thank many persons. The number of colleagues and friends I<br />
have learned from about mathematical physics and especially disordered systems<br />
is so large that it is impossible <strong>to</strong> mention a few without do<strong>in</strong>g <strong>in</strong>justice <strong>to</strong> many<br />
others. A lot of the names can be found as authors <strong>in</strong> the list of references. Without<br />
these persons the writ<strong>in</strong>g of this review would have been impossible.<br />
A colleague and friend I have <strong>to</strong> mention though is Frédéric Klopp who organized<br />
a summer school on <strong>Random</strong> Schröd<strong>in</strong>ger opera<strong>to</strong>rs <strong>in</strong> Paris <strong>in</strong> 2002. My lectures<br />
there were the start<strong>in</strong>g po<strong>in</strong>t for this review. I have <strong>to</strong> thank Frédéric especially<br />
for his enormous patience when I did not obey the third, forth, . . . , deadl<strong>in</strong>e for<br />
deliver<strong>in</strong>g the manuscript.<br />
It is a great pleasure <strong>to</strong> thank Bernd Metzger for his advice, for many helpful discussions,<br />
for proofread<strong>in</strong>g the manuscript, for help<strong>in</strong>g me with the text and especially<br />
with the references and for many other th<strong>in</strong>gs.<br />
Last, not least I would like <strong>to</strong> thank Jessica Langner, Riccardo Catalano and Hendrik<br />
Meier for the skillful typ<strong>in</strong>g of the manuscript, for proofread<strong>in</strong>g and for their<br />
patience with the author.<br />
Notes and Remarks<br />
For the spectral theory needed <strong>in</strong> this work we recommend [117] or [141]. We will<br />
also need the m<strong>in</strong>-max theorem (see [115]).<br />
The probabilistic background we need can be found e.g. <strong>in</strong> [95] and [96].<br />
For further read<strong>in</strong>g on random Schröd<strong>in</strong>ger opera<strong>to</strong>rs we recommend [78] for the<br />
state of the art <strong>in</strong> multiscale analysis. We also recommend the textbook [128]. A<br />
modern survey on the density of states is [67].
7<br />
2. Introduction: Why random Schröd<strong>in</strong>ger opera<strong>to</strong>rs <br />
2.1. The sett<strong>in</strong>g of quantum mechanics.<br />
A quantum mechanical particle mov<strong>in</strong>g <strong>in</strong> d-dimensional space is described by a<br />
vec<strong>to</strong>r ψ <strong>in</strong> the Hilbert space L 2 (R d ). The time evolution of the state ψ is determ<strong>in</strong>ed<br />
by the Schröd<strong>in</strong>ger opera<strong>to</strong>r<br />
H = H 0 + V (2.1)<br />
act<strong>in</strong>g on L 2 (R d ). The opera<strong>to</strong>r H 0 is called the free opera<strong>to</strong>r. It represents the<br />
k<strong>in</strong>etic energy of the particle. In the absence of magnetic fields it is given by the<br />
Laplacian<br />
H 0 = − 2<br />
2m ∆ = − 2<br />
2m<br />
d∑<br />
ν=1<br />
∂ 2<br />
∂x ν<br />
2 . (2.2)<br />
The physics of the system is encoded <strong>in</strong> the potential V which is the multiplication<br />
opera<strong>to</strong>r with the function V (x) <strong>in</strong> the Hilbert space L 2 (R d ). The function V (x)<br />
is the (classical) potential energy. Consequently, the forces are given by<br />
F (x) = −∇V (x) .<br />
<br />
In the follow<strong>in</strong>g we choose physical units <strong>in</strong> such a way that 2<br />
2m<br />
= 1 s<strong>in</strong>ce we<br />
are not <strong>in</strong>terested <strong>in</strong> the explicit dependence of quantities on or m. The time<br />
evolution of the state ψ is obta<strong>in</strong>ed from the time dependent Schröd<strong>in</strong>ger equation<br />
i ∂ ∂t ψ = H ψ . (2.3)<br />
By the spectral theorem for self adjo<strong>in</strong>t opera<strong>to</strong>rs equation (2.3) can be solved by<br />
ψ(t) = e −itH ψ 0 (2.4)<br />
where ψ 0 is the state of the system at time t = 0.<br />
To extract valuable <strong>in</strong>formation from (2.4) we have <strong>to</strong> know as much as possible<br />
about the spectral theory of the opera<strong>to</strong>r H and this is what we try <strong>to</strong> do <strong>in</strong> this text.<br />
2.2. <strong>Random</strong> Potentials.<br />
In this review we are <strong>in</strong>terested <strong>in</strong> random Schröd<strong>in</strong>ger opera<strong>to</strong>rs. These opera<strong>to</strong>rs<br />
model disordered solids. Solids occur <strong>in</strong> nature <strong>in</strong> various forms. Sometimes they<br />
are (almost) <strong>to</strong>tally ordered. In crystals the a<strong>to</strong>ms or nuclei are distributed on a<br />
periodic lattice (say the lattice Z d for simplicity) <strong>in</strong> a completely regular way. Let<br />
us assume that a particle (electron) at the po<strong>in</strong>t x ∈ R d feels a potential of the form<br />
q f(x − i) due <strong>to</strong> an a<strong>to</strong>m (or ion or nucleus) located at the po<strong>in</strong>t i ∈ Z d . Here, the<br />
constant q, the charge or coupl<strong>in</strong>g constant <strong>in</strong> physical terms, could be absorbed<br />
<strong>in</strong><strong>to</strong> the function f. However, s<strong>in</strong>ce we are go<strong>in</strong>g <strong>to</strong> vary this quantity from a<strong>to</strong>m <strong>to</strong><br />
a<strong>to</strong>m later on, it is useful <strong>to</strong> write the potential <strong>in</strong> the above way. Then, <strong>in</strong> a regular<br />
crystal our particle is exposed <strong>to</strong> a <strong>to</strong>tal potential
8<br />
V (x) = ∑ i∈Z d q f(x − i) . (2.5)<br />
We call the function f the s<strong>in</strong>gle site potential <strong>to</strong> dist<strong>in</strong>guish it from the <strong>to</strong>tal<br />
potential V . The potential V <strong>in</strong> (2.5) is periodic with respect <strong>to</strong> the lattice Z d ,<br />
i. e. V (x − i) = V (x) for all x ∈ R d and i ∈ Z d . The mathematical theory<br />
of Schröd<strong>in</strong>ger opera<strong>to</strong>rs with periodic potentials is well developed (see e.g. [41],<br />
[115] ). It is based on a thorough analysis of the symmetry properties of periodic<br />
opera<strong>to</strong>rs. For example, it is known that such opera<strong>to</strong>rs have a spectrum with band<br />
structure, i.e. σ(H) = ⋃ ∞<br />
n=0 [a n, b n ] with a n < b n ≤ a n+1 . This spectrum is also<br />
known <strong>to</strong> be absolutely cont<strong>in</strong>uous.<br />
Most solids do not constitute ideal crystals. The positions of the a<strong>to</strong>ms may deviate<br />
from the ideal lattice positions <strong>in</strong> a non regular way due <strong>to</strong> imperfections <strong>in</strong><br />
the crystallization process. Or the positions of the a<strong>to</strong>ms may be completely disordered<br />
as is the case <strong>in</strong> amorphous or glassy materials. The solid may also be<br />
a mixture of various materials which is the case for example for alloys or doped<br />
semiconduc<strong>to</strong>rs. In all these cases it seems reasonable <strong>to</strong> look upon the potential<br />
as a random quantity.<br />
For example, if the material is a pure one, but the positions of the a<strong>to</strong>ms deviate<br />
from the ideal lattice positions randomly, we may consider a random potential of<br />
the form<br />
V ω (x) = ∑ i∈Z d q f (x − i − ξ i (ω)) . (2.6)<br />
Here the ξ i are random variables which describe the deviation of the ‘i th ’ a<strong>to</strong>m<br />
from the lattice position i. One may, for example assume that the random variables<br />
ξ i are <strong>in</strong>dependent and identically distributed. We have added a subscript ω <strong>to</strong> the<br />
potential V <strong>to</strong> make clear that V ω depends on (unknown) random parameters.<br />
To model an amorphous material like glass or rubber we assume that the a<strong>to</strong>ms of<br />
the material are located at completely random po<strong>in</strong>ts η i <strong>in</strong> space. Such a random<br />
potential may formally be written as<br />
V ω (x) = ∑ i∈Z d q f(x − η i ). (2.7)<br />
To write the potential (2.7) as a sum over the lattice Z d is somewhat mislead<strong>in</strong>g,<br />
s<strong>in</strong>ce there is, <strong>in</strong> general, no natural association of the η i with a lattice po<strong>in</strong>t i. It is<br />
more appropriate <strong>to</strong> th<strong>in</strong>k of a collection of random po<strong>in</strong>ts <strong>in</strong> R d as a random po<strong>in</strong>t<br />
measure. This representation emphasizes that any order<strong>in</strong>g of the η i is completely<br />
artificial.<br />
A count<strong>in</strong>g measure is a Borel measure on R d of the form ν = ∑ x∈M δ x with a<br />
countable set M without (f<strong>in</strong>ite) accumulation po<strong>in</strong>ts. By a random po<strong>in</strong>t measure<br />
we mean a mapp<strong>in</strong>g ω ↦→ µ ω , such that µ ω is a count<strong>in</strong>g measure with the property<br />
that the function ω ↦→ µ ω (A) is measurable for any bounded Borel set A. If ν = ν ω
is the random po<strong>in</strong>t measure ν = ∑ i δ η i<br />
then (2.7) can be written as<br />
∫<br />
V ω (x) = q f(x − η) dν(η) . (2.8)<br />
R d<br />
The most frequently used example of a random po<strong>in</strong>t measure and the most important<br />
one is the Poisson random measure µ ω . Let us set n A = µ ω (A), the number<br />
of random po<strong>in</strong>ts <strong>in</strong> the set A. The Poisson random measure can be characterized<br />
by the follow<strong>in</strong>g specifications<br />
• The random variables n A and n B are <strong>in</strong>dependent for disjo<strong>in</strong>t (measurable)<br />
sets A and B.<br />
• The probability that n A<br />
Lebesgue measure of A.<br />
= k is equal <strong>to</strong> |A|k<br />
k!<br />
e −|A| , where |A| is the<br />
A random potential of the form (2.8) with the Poisson random measure is called<br />
the Poisson model .<br />
The most popular model of a disordered solid and the best unders<strong>to</strong>od one as well<br />
is the alloy-type potential (see (2.9) below). It models an unordered alloy, i.e.<br />
a mixture of several materials the a<strong>to</strong>ms of which are located at lattice positions.<br />
The type of a<strong>to</strong>m at the lattice po<strong>in</strong>t i is assumed <strong>to</strong> be random. In the model we<br />
consider here the different materials are described by different charges (or coupl<strong>in</strong>g<br />
constants) q i . The <strong>to</strong>tal potential V is then given by<br />
V ω (x) = ∑ i∈Z d q i (ω) f(x − i) . (2.9)<br />
The q i are random variables which we assume <strong>to</strong> be <strong>in</strong>dependent and identically<br />
distributed. Their range describes the possible values the coupl<strong>in</strong>g constant can<br />
assume <strong>in</strong> the considered alloy. The physical model suggests that there are only<br />
f<strong>in</strong>itely many values the random variables can assume. However, <strong>in</strong> the proofs of<br />
some results we have <strong>to</strong> assume that the distribution of the random variables q i<br />
is cont<strong>in</strong>uous (even absolutely cont<strong>in</strong>uous) due <strong>to</strong> limitations of the mathematical<br />
techniques. One might argue that such an assumption is acceptable as a purely<br />
technical one. On the other hand one could say we have not unders<strong>to</strong>od the problem<br />
as long as we can not handle the physically relevant cases.<br />
For a given ω the potential V ω (x) is a pretty complicated ‘normal’ function. So,<br />
one may ask: What is the advantage of ‘mak<strong>in</strong>g it random’<br />
With the <strong>in</strong>troduction of random variables we implicitly change our po<strong>in</strong>t of view.<br />
From now on we are hardly <strong>in</strong>terested <strong>in</strong> properties of H ω for a s<strong>in</strong>gle given ω.<br />
Rather, we look at ‘typical’ properties of H ω . In mathematical terms, we are <strong>in</strong>terested<br />
<strong>in</strong> results of the form: The set of all ω such that H ω has the property P has<br />
probability one. In short: P holds for P-almost all ω (or P-almost surely). Here P<br />
is the probability measure on the underly<strong>in</strong>g probability space.<br />
In this course we will encounter a number of such properties. For example we will<br />
see that (under weak assumptions on V ω ), there is a closed, nonrandom (!) subset<br />
Σ of the real l<strong>in</strong>e such that Σ = σ(H ω ), the spectrum of the opera<strong>to</strong>r H ω , P-almost<br />
surely.<br />
9
10<br />
This and many other results can be proven for various types of random Schröd<strong>in</strong>ger<br />
opera<strong>to</strong>rs. In this lecture we will restrict ourselves <strong>to</strong> a relatively simple system<br />
known as the <strong>An</strong>derson model. Here the Hilbert space is the sequence space<br />
l 2 (Z d ) <strong>in</strong>stead of L 2 (R d ) and the free opera<strong>to</strong>r H 0 is a f<strong>in</strong>ite-difference opera<strong>to</strong>r<br />
rather than the Laplacian. We will call this sett<strong>in</strong>g the discrete case <strong>in</strong> contrast<br />
<strong>to</strong> Schröd<strong>in</strong>ger opera<strong>to</strong>rs on L 2 (R d ) which we refer <strong>to</strong> as the cont<strong>in</strong>uous case . In<br />
the references the reader may f<strong>in</strong>d papers which extend results we prove here <strong>to</strong> the<br />
cont<strong>in</strong>uous sett<strong>in</strong>g.<br />
2.3. The one body approximation.<br />
In the above sett<strong>in</strong>g we have implicitly assumed that we describe a s<strong>in</strong>gle particle<br />
mov<strong>in</strong>g <strong>in</strong> a static exterior potential. This is at best a caricature of what we f<strong>in</strong>d <strong>in</strong><br />
nature. First of all there are many electrons mov<strong>in</strong>g <strong>in</strong> a solid and they <strong>in</strong>teract with<br />
each other. The exterior potential orig<strong>in</strong>ates <strong>in</strong> nuclei or ions which are themselves<br />
<strong>in</strong>fluenced both by the other nuclei and by the electrons. In the above discussion<br />
we have also implicitly assumed that the solid we consider extends <strong>to</strong> <strong>in</strong>f<strong>in</strong>ity <strong>in</strong> all<br />
directions, (i.e. fills the universe). Consequently, we ought <strong>to</strong> consider <strong>in</strong>f<strong>in</strong>itely<br />
many <strong>in</strong>teract<strong>in</strong>g particles. It is obvious that such a task is out of range of the<br />
methods available <strong>to</strong>day. As a first approximation it seems quite reasonable <strong>to</strong><br />
separate the motion of the nuclei from the system and <strong>to</strong> take the nuclei <strong>in</strong><strong>to</strong> account<br />
only via an exterior potential. Indeed, the masses of the nuclei are much larger than<br />
those of the electrons.<br />
The second approximation is <strong>to</strong> neglect the electron-electron <strong>in</strong>teraction. It is not at<br />
all clear that this approximation gives a qualitatively correct picture. In fact, there<br />
is physical evidence that the <strong>in</strong>teraction between the electrons is fundamental for a<br />
number of phenomena.<br />
Interact<strong>in</strong>g particle systems <strong>in</strong> condensed matter are an object of <strong>in</strong>tensive research<br />
<strong>in</strong> theoretical physics. In mathematics, however, this field of research is still <strong>in</strong> its<br />
<strong>in</strong>fancy despite of an <strong>in</strong>creas<strong>in</strong>g <strong>in</strong>terest <strong>in</strong> the subject.<br />
If we neglect the <strong>in</strong>teractions between the electrons we are left with a system of<br />
non<strong>in</strong>teract<strong>in</strong>g electrons <strong>in</strong> an exterior potential. It is not hard <strong>to</strong> see that such<br />
a system (and the correspond<strong>in</strong>g Hamil<strong>to</strong>nian) separates, i.e. the eigenvalues are<br />
just sums of the one-body eigenvalues and the eigenfunctions have product form.<br />
So, if ψ 1 , ψ 2 , . . . , ψ N are eigenfunctions of the one-body system correspond<strong>in</strong>g <strong>to</strong><br />
eigenvalues E 1 , E 2 , . . . , E N respectively, then<br />
Ψ(x 1 , x 2 , . . . , x N ) = ψ 1 (x 1 ) · ψ 2 (x 2 ) · . . . · ψ N (x N ) . (2.10)<br />
is an eigenfunction of the full system with eigenvalue E 1 + E 2 + . . . + E N .<br />
However, there is a subtlety <strong>to</strong> obey here, which is typical <strong>to</strong> many particle Quantum<br />
Mechanics. The electrons <strong>in</strong> the solid are <strong>in</strong>dist<strong>in</strong>guishable, s<strong>in</strong>ce we are unable<br />
<strong>to</strong> ‘follow their trajec<strong>to</strong>ries’. The correspond<strong>in</strong>g Hamil<strong>to</strong>nian is <strong>in</strong>variant under<br />
permutation of the particles. As a consequence, the N-particle Hilbert space<br />
consists either of <strong>to</strong>tally symmetric or of <strong>to</strong>tally antisymmetric functions of the particle<br />
positions x 1 , x 2 , . . . , x N . It turns out that for particles with <strong>in</strong>teger sp<strong>in</strong> the
symmetric subspace is the correct one. Such particles, like pho<strong>to</strong>ns, phonons or<br />
mesons, are called Bosons.<br />
Electrons, like pro<strong>to</strong>ns and neutrons, are Fermions, particles with half <strong>in</strong>teger sp<strong>in</strong>.<br />
The Hilbert space for Fermions consists of <strong>to</strong>tally antisymmetric functions, i.e.:<br />
if x 1 , x 2 , . . . , x N ∈ R d are the coord<strong>in</strong>ates of N electrons, then any state ψ of the<br />
system satisfies ψ(x 1 , x 2 , x 3 , . . . , x N ) = −ψ(x 2 , x 1 , x 3 , . . . , x N ) and similarly<br />
for <strong>in</strong>terchang<strong>in</strong>g any other pair of particles.<br />
It follows, that the product <strong>in</strong> (2.10) is not a vec<strong>to</strong>r of the (correct) Hilbert space<br />
(of antisymmetric functions). Only its anti-symmetrization is<br />
Ψ f (x 1 , x 2 , . . . , x N ) := ∑<br />
π∈S N<br />
(−1) π ψ 1 (x π1 )ψ 2 (x π2 ) . . . , ψ N (x πN ) . (2.11)<br />
Here, the symbol S N stands for the permutation group and (−1) π equals 1 for even<br />
permutations (i.e. products of an even number of exchanges), it equals −1 for odd<br />
permutations.<br />
The anti-symmetrization (2.11) is non zero only if the functions ψ j are pairwise<br />
different. Consequently, the eigenvalues of the multi-particle system are given as<br />
sums E 1 + E 2 + . . . + E N of the eigenvalues E j of the one-particle system where<br />
the eigenvalues E j are all different. (We count multiplicity, i.e. an eigenvalue of<br />
multiplicity two may occur twice <strong>in</strong> the above sum). This rule is known as the<br />
Pauli-pr<strong>in</strong>ciple.<br />
The ground state energy of a system of N identical, non<strong>in</strong>teract<strong>in</strong>g Fermions is<br />
therefore given by<br />
11<br />
E 1 + E 2 + . . . + E N<br />
where the E n are the eigenvalues of the s<strong>in</strong>gle particle system <strong>in</strong> <strong>in</strong>creas<strong>in</strong>g order,<br />
E 1 ≤ E 2 ≤ . . . counted accord<strong>in</strong>g <strong>to</strong> multiplicity.<br />
It is not at all obvious how we can implement the above rules for the systems<br />
considered here. Their spectra tend <strong>to</strong> consist of whole <strong>in</strong>tervals rather than be<strong>in</strong>g<br />
discrete, moreover, s<strong>in</strong>ce the systems extend <strong>to</strong> <strong>in</strong>f<strong>in</strong>ity they ought <strong>to</strong> have <strong>in</strong>f<strong>in</strong>itely<br />
many electrons.<br />
To circumvent this difficulty we will <strong>in</strong>troduce a procedure known as the ‘thermodynamic<br />
limit’: We first restrict the system <strong>to</strong> a f<strong>in</strong>ite but large box (of length L<br />
say), then we def<strong>in</strong>e quantities of <strong>in</strong>terest <strong>in</strong> this system, for example the number<br />
of states <strong>in</strong> a given energy region per unit volume. F<strong>in</strong>ally, we let the box grow <strong>in</strong>def<strong>in</strong>itely<br />
(i.e. send L <strong>to</strong> <strong>in</strong>f<strong>in</strong>ity) and hope (or better prove) that the quantity under<br />
consideration has a limit as L goes <strong>to</strong> <strong>in</strong>f<strong>in</strong>ity. In the case of the number of states<br />
per unit volume this limit, <strong>in</strong> deed, exists. It is called the density of states measure<br />
and will play a major role <strong>in</strong> what follows. We will discuss this issue <strong>in</strong> detail <strong>in</strong><br />
chapter 5.
12<br />
Notes and Remarks<br />
Standard references for mathematical methods of quantum mechanics are [57],<br />
[117], [114], [116], [115] and [141], [30].<br />
Most of the necessary prerequisites from spectral theory can be found <strong>in</strong> [117] or<br />
[141]. A good source for the probabilistic background is [95] and [96].<br />
The physical theory of random Schröd<strong>in</strong>ger opera<strong>to</strong>rs is described <strong>in</strong> [9], [99],<br />
[135] and [136]. References for the mathematical approach <strong>to</strong> random Schröd<strong>in</strong>ger<br />
opera<strong>to</strong>rs are [23], [30], [97], [58], [112] and [128].
13<br />
3. Setup: The <strong>An</strong>derson model<br />
3.1. Discrete Schröd<strong>in</strong>ger opera<strong>to</strong>rs.<br />
In the <strong>An</strong>derson model the Hilbert space L 2 (R d ) is replaced by the sequence space<br />
l 2 (Z d ) = {(u i ) i∈Z d| ∑ |u i | 2 < ∞} (3.1)<br />
i∈Z d<br />
= {u : Z d → C ∣ ∣ ∑ n∈Z d |u(n)| 2 < ∞} . (3.2)<br />
We denote the norm on l 2 (Z d ) by<br />
|| u|| =<br />
( ∑<br />
n∈Z d |u(n)| 2 ) 1 2<br />
(3.3)<br />
Here, we th<strong>in</strong>k of a particle mov<strong>in</strong>g on the lattice Z d , so that <strong>in</strong> the case ‖u‖ = 1<br />
the probability <strong>to</strong> f<strong>in</strong>d the particle at the po<strong>in</strong>t n ∈ Z d is given by |u(n)| 2 . Note,<br />
that we may th<strong>in</strong>k of u either as a function u(n) on Z d or as a sequence u n <strong>in</strong>dexed<br />
by Z d .<br />
It will be convenient <strong>to</strong> equip Z d with two different norms. The first one is<br />
|| n || ∞ := sup | n ν | . (3.4)<br />
ν=1,...,d<br />
This norm respects the cubic structure of the lattice Z d . For example, it is convenient<br />
<strong>to</strong> def<strong>in</strong>e the cubes (n 0 ∈ Z d , L ∈ N)<br />
Λ L (n 0 ) := {n ∈ Z d ; || n − n 0 || ∞ ≤ L} . (3.5)<br />
Λ L (n 0 ) is the cube of side length 2L + 1 centered at n 0 . It conta<strong>in</strong>s |Λ L (n 0 ) | :=<br />
(2L + 1) d po<strong>in</strong>ts. Sometimes we call |Λ L (n 0 ) | the volume of Λ L (n 0 ). In general,<br />
we denote by |A| the number of elements of the set A. To shorten notation we<br />
write Λ L for Λ L (0). The other norm we use on Z d is<br />
|| n|| 1 :=<br />
d∑<br />
| n ν | . (3.6)<br />
ν=1<br />
This norm reflects the graph structure of Z d . Two vertices n and m of the graph<br />
Z d are connected by an edge, if they are nearest neighbors, i. e. if || n − m|| 1 = 1.<br />
For arbitrary n, m ∈ Z d the norm || n − m|| 1 gives the length of the shortest path<br />
between n and m.<br />
The k<strong>in</strong>etic energy opera<strong>to</strong>r H 0 is a discrete analogue of the (negative) Laplacian,<br />
namely
14<br />
(H 0 u)(n) = − ∑<br />
|| m−n|| 1 =1<br />
(u(m) − u(n)) . (3.7)<br />
This opera<strong>to</strong>r is also known as the graph Laplacian for the graph Z d or the discrete<br />
Laplacian. Its quadratic form is given by<br />
〈u, H 0 v〉 = 1 ∑ ∑<br />
(u (n) − u (m)) (v(n) − v(m)) . (3.8)<br />
2<br />
n∈Z d<br />
|| m−n|| 1 =1<br />
We call this sesquil<strong>in</strong>ear form a Dirichlet form because of its similarity <strong>to</strong> the classical<br />
Dirichlet form<br />
∫<br />
〈u, −△ v〉 = ∇u(x) · ∇v(x) dx .<br />
R d<br />
The opera<strong>to</strong>r H 0 is easily seen <strong>to</strong> be symmetric and bounded, <strong>in</strong> fact<br />
( ∑ ( ∑<br />
‖H 0 u‖ =<br />
n∈Z d<br />
|| j || 1 =1<br />
(<br />
u(n + j) − u(n)<br />
) ) 2) 1<br />
2<br />
(3.9)<br />
≤<br />
∑ ( ∑<br />
| u(n + j) − u(n) | 2) 1 2<br />
(3.10)<br />
|| j || 1 =1 n∈Z d<br />
≤<br />
∑ ( ∑ ( ) ) 1<br />
2 2<br />
| u(n + j) | + | u(n) | (3.11)<br />
|| j|| 1 =1 n∈Z d<br />
≤<br />
∑ ( ( ∑<br />
) 1 ( ∑<br />
| u(n + j) | 2 2<br />
+ | u(n) | 2) )<br />
1<br />
2<br />
(3.12)<br />
|| j|| 1 =1 n∈Z d n∈Z d<br />
≤ 4 d ‖ u ‖ . (3.13)<br />
From l<strong>in</strong>e (3.9) <strong>to</strong> (3.10) we applied the triangle <strong>in</strong>equality for l 2 <strong>to</strong> the functions<br />
∑<br />
fj (n) with f j (n) = u(n + j) − u(n). In (3.12) and (3.13) we used the triangle<br />
<strong>in</strong>equality and the fact that any lattice po<strong>in</strong>t <strong>in</strong> Z d has 2d neighbors.<br />
Let us def<strong>in</strong>e the Fourier transform from l 2 (Z d ) <strong>to</strong> L 2 ([0, 2π] d ) by<br />
(Fu)(k) = û(k) = ∑ n<br />
u n e −i n·k . (3.14)<br />
F is a unitary opera<strong>to</strong>r. Under F the discrete Laplacian H 0 transforms <strong>to</strong> the<br />
multiplication opera<strong>to</strong>r with the function h 0 (k) = 2 ∑ d<br />
ν=1 (1 − cos(k ν)), i.e.<br />
FH 0 F −1 is the multiplication opera<strong>to</strong>r on L 2 ([0, 2π] d ) with the function h 0 . This<br />
shows that the spectrum σ(H 0 ) equals [0, 4d] (the range of the function h 0 ) and<br />
that H 0 has purely absolutely cont<strong>in</strong>uous spectrum.<br />
It is very convenient that the ‘discrete Dirac function’ δ i def<strong>in</strong>ed by (δ i ) j = 0 for<br />
i ≠ j and (δ i ) i = 1 is an ‘honest’ l 2 -vec<strong>to</strong>r, <strong>in</strong> fact the collection {δ i } i∈Z d is an
orthonormal basis of l 2 (Z d ). This allows us <strong>to</strong> def<strong>in</strong>e matrix entries or a kernel for<br />
every (say bounded) opera<strong>to</strong>r A on l 2 (Z d ) by<br />
A(i, j) = 〈δ i , Aδ j 〉 . (3.15)<br />
15<br />
We have (Au)(i) = ∑ j∈Zd A(i, j)u(j). So, the A(i, j) def<strong>in</strong>e the opera<strong>to</strong>r A<br />
uniquely.<br />
In this representation the multiplication opera<strong>to</strong>r V is diagonal, while<br />
⎧<br />
⎨<br />
H 0 (i, j) =<br />
⎩<br />
−1 if || i − j|| 1 = 1,<br />
2d if i = j,<br />
0 otherwise.<br />
(3.16)<br />
In many texts the diagonal term <strong>in</strong> H 0 is dropped and absorbed <strong>in</strong><strong>to</strong> the potential<br />
V . Moreover, one can also neglect the −-sign <strong>in</strong> the offdiagonal terms of (3.16).<br />
The correspond<strong>in</strong>g opera<strong>to</strong>r is up <strong>to</strong> a constant equivalent <strong>to</strong> H 0 and has spectrum<br />
[−2d, 2d].<br />
In this sett<strong>in</strong>g the potential V is a multiplication opera<strong>to</strong>r with a function V (n) on<br />
Z d . The simplest form <strong>to</strong> make this random is <strong>to</strong> take V (n) = V ω (n) itself as<br />
<strong>in</strong>dependent, identically distributed random variables (see Section 3.4), so we have<br />
H ω = H 0 + V ω .<br />
We call this random opera<strong>to</strong>r the <strong>An</strong>derson model . For most of this course we will<br />
be concerned with this opera<strong>to</strong>r.<br />
3.2. Spectral calculus.<br />
One of the most important <strong>to</strong>ols of spectral theory is the functional calculus (spectral<br />
theorem) for self adjo<strong>in</strong>t opera<strong>to</strong>rs. We discuss this <strong>to</strong>pic here by giv<strong>in</strong>g a brief<br />
sketch of the theory and establish notations. Details about functional calculus can<br />
be found <strong>in</strong> [117]. (For an alternative approach see [34]).<br />
Throughout this section let A denote a self adjo<strong>in</strong>t opera<strong>to</strong>r with doma<strong>in</strong> D(A) on<br />
a (separable) Hilbert space H. We will try <strong>to</strong> def<strong>in</strong>e functions f(A) of A for a huge<br />
class of functions f. Some elementary functions of A can be def<strong>in</strong>ed <strong>in</strong> an obvious<br />
way. One example is the resolvent which we consider first.<br />
For any z ∈ C the opera<strong>to</strong>r A − z = A − z id is def<strong>in</strong>ed by (A − z)ϕ =<br />
Aϕ − zϕ. The resolvent set ρ(A) of A is the set of all z ∈ C for which A − z<br />
is a bijective mapp<strong>in</strong>g from D(A) <strong>to</strong> H. The spectrum σ(A) of A is def<strong>in</strong>ed by<br />
σ(A) = C \ ρ(A). For self adjo<strong>in</strong>t A we have σ(A) ⊂ R. The spectrum is always<br />
a closed set. If A is bounded, σ(A) is compact.<br />
For z ∈ ρ(A) we can <strong>in</strong>vert A − z. The <strong>in</strong>verse opera<strong>to</strong>r (A − z) −1 is called the<br />
resolvent of A. For self adjo<strong>in</strong>t A the (A − z) −1 is a bounded opera<strong>to</strong>r for all<br />
z ∈ ρ(A).<br />
Resolvents observe the follow<strong>in</strong>g important identities, known as the resolvent equations
16<br />
(A − z 1 ) −1 − (A − z 2 ) −1 = (z 1 − z 2 ) (A − z 1 ) −1 (A − z 2 ) −1 (3.17)<br />
and, if D(A) = D(B),<br />
= (z 1 − z 2 ) (A − z 2 ) −1 (A − z 1 ) −1 (3.18)<br />
(A − z) −1 − (B − z) −1 = (A − z) −1 (B − A) (B − z) −1 (3.19)<br />
For z ∈ C and M ⊂ C we def<strong>in</strong>e<br />
= (B − z) −1 (B − A) (A − z) −1 (3.20)<br />
dist(z, M) = <strong>in</strong>f{|z − ζ|; ζ ∈ M} (3.21)<br />
It is not hard <strong>to</strong> see that for any self adjo<strong>in</strong>t opera<strong>to</strong>r A and any z ∈ ρ(A) the<br />
opera<strong>to</strong>r norm ‖(A − z) −1 ‖ of the resolvent is given by<br />
‖(A − z) −1 ‖ =<br />
1<br />
dist(z, σ(A)) . (3.22)<br />
In particular, for a self adjo<strong>in</strong>t opera<strong>to</strong>r A and z ∈ C \ R<br />
‖(A − z) −1 ‖ ≤<br />
1<br />
Im z . (3.23)<br />
For the rest of this section we assume that the opera<strong>to</strong>r A is bounded. In this case,<br />
polynomials of the opera<strong>to</strong>r A can be def<strong>in</strong>ed straightforwardly<br />
A 2 ϕ = A ( A(ϕ) ) (3.24)<br />
(<br />
A 3 ϕ = A A ( A(Aϕ) )) etc. (3.25)<br />
More generally, if P is a complex valued polynomial <strong>in</strong> one real variable, P (λ) =<br />
∑ n<br />
j=0 a nλ j then<br />
It is a key observation that<br />
P (A) =<br />
n∑<br />
a n A j . (3.26)<br />
j=0<br />
‖P (A)‖ =<br />
sup<br />
λ∈σ(A)<br />
|P (λ)| (3.27)<br />
Let now f be a function <strong>in</strong> C ( σ(A) ) the complex-valued cont<strong>in</strong>uous functions on<br />
(the compact set) σ(A). The Weierstraß approximation theorem tells us, that on<br />
σ(A) the function f can be uniformly approximated by polynomials. Thus us<strong>in</strong>g<br />
(3.27) we can def<strong>in</strong>e the opera<strong>to</strong>r f(A) as a norm limit of polynomials P n (A).<br />
These opera<strong>to</strong>rs satisfy
17<br />
(αf + βg) (A) = αf(A) + βg(A) (3.28)<br />
f · g (A) = f(A) g(A) (3.29)<br />
f (A) = f(A) ∗ (3.30)<br />
If f ≥ 0 then 〈ϕ, f(A)ϕ〉 ≥ 0 for all ϕ ∈ H (3.31)<br />
By the Riesz-representation theorem it follows, that for each ϕ ∈ H there is a<br />
positive and bounded measure µ ϕ,ϕ on σ(A) such that for all f ∈ C ( σ(A) )<br />
〈ϕ, f(A)ϕ〉 =<br />
∫<br />
f(λ) dµ ϕ,ϕ (λ) . (3.32)<br />
For ϕ, ψ ∈ H, us<strong>in</strong>g the polarization identity, we f<strong>in</strong>d complex-valued measures<br />
µ ϕ,ψ such that<br />
〈ϕ, f(A)ϕ〉 =<br />
∫<br />
f(λ) dµ ϕ,ψ (λ) . (3.33)<br />
Equation (3.33) can be used <strong>to</strong> def<strong>in</strong>e the opera<strong>to</strong>r f(A) for bounded measurable<br />
functions. The opera<strong>to</strong>rs f(A), g(A) satisfy (3.28)–(3.31) for bounded measurable<br />
functions as well, moreover we have:<br />
‖f(A)‖ ≤ sup |f(λ)| (3.34)<br />
λ∈σ(A)<br />
with equality for cont<strong>in</strong>uous f.<br />
For any Borel set M ⊂ R we denote by χ M the characteristic function of M<br />
def<strong>in</strong>ed by<br />
χ M (λ) =<br />
{<br />
1 if λ ∈ M<br />
0 otherwise.<br />
(3.35)<br />
The opera<strong>to</strong>rs µ(A) = χ M (A) play a special role. It is not hard <strong>to</strong> check that they<br />
satisfy the follow<strong>in</strong>g conditions:<br />
µ(A) is an orthogonal projection. (3.36)<br />
µ(∅) = 0 and µ ( σ(A) ) = 1 (3.37)<br />
µ(M ∩ N) = µ(M) µ(N) (3.38)<br />
If the Borel sets M n are pairwise disjo<strong>in</strong>t, then for each ϕ ∈ H<br />
µ ( ∞ ⋃<br />
n=1<br />
M n<br />
)<br />
ϕ =<br />
∞ ∑<br />
n=1<br />
µ(M n ) ϕ (3.39)<br />
S<strong>in</strong>ce µ(M) = χ M (A) satisfies (3.36)–(3.39) it is called the projection valued<br />
measure associated <strong>to</strong> the opera<strong>to</strong>r A or the projection valued spectral measure of<br />
A. We have
18<br />
〈ϕ, µ(M)ψ〉 = µ ϕ,ψ (M) (3.40)<br />
The functional calculus can be implemented for unbounded self adjo<strong>in</strong>t opera<strong>to</strong>rs<br />
as well. For such opera<strong>to</strong>rs the spectrum is always a closed set. It is compact only<br />
for bounded opera<strong>to</strong>rs.<br />
We will use the functional calculus virtually everywhere throughout this paper. For<br />
example, it gives mean<strong>in</strong>g <strong>to</strong> the opera<strong>to</strong>r e −itH used <strong>in</strong> (2.4). We will look at the<br />
projection valued measures χ M (A) more closely <strong>in</strong> chapter 7.<br />
3.3. Some more functional analysis.<br />
In this section we recall a few results from functional analysis and spectral theory<br />
and establish notations at the same time. In particular, we discuss the m<strong>in</strong>-max<br />
pr<strong>in</strong>ciple and the S<strong>to</strong>ne-Weierstraß theorem.<br />
Let A be a selfadjo<strong>in</strong>t (not necessarily bounded) opera<strong>to</strong>r on the (separable) Hilbert<br />
space H with doma<strong>in</strong> D(A). We denote the set of eigenvalues of A by ε(A). Obviously,<br />
any eigenvalue of A belongs <strong>to</strong> the spectrum σ(A). The multiplicity of an<br />
eigenvalue λ of A is the dimension of the eigenspace {ϕ ∈ D(A); Aϕ = λϕ}<br />
associated <strong>to</strong> λ. If µ is the projection valued spectral measure of A, then the<br />
multiplicity of λ equals tr µ({λ}). <strong>An</strong> eigenvalue is called simple or non degenerate<br />
if its multiplicity is one, it is called f<strong>in</strong>itely degenerate if its eigenspace<br />
is f<strong>in</strong>ite dimensional. <strong>An</strong> eigenvalue λ is called isolated if there is an ε > 0<br />
such that σ(A) ∩ (λ − ε, λ + ε) = {λ}. <strong>An</strong>y isolated po<strong>in</strong>t <strong>in</strong> the spectrum<br />
is always an eigenvalue. The discrete spectrum σ dis (A) is the set of all isolated<br />
eigenvalues of f<strong>in</strong>ite multiplicity. The essential spectrum σ ess (A) is def<strong>in</strong>ed by<br />
σ ess (A) = σ(A) \ σ dis (A).<br />
The opera<strong>to</strong>r A is called positive if 〈φ, Aφ〉 ≥ 0 for all φ <strong>in</strong> the doma<strong>in</strong> D(A), A<br />
is called bounded below if 〈φ, Aφ〉 ≥ −M〈φ, φ〉 for some M and all φ ∈ D(A).<br />
We def<strong>in</strong>e<br />
µ 0 (A) = <strong>in</strong>f {〈φ, Aφ〉 ; φ ∈ D(A), ‖φ‖ = 1} (3.41)<br />
and for k ≥ 1<br />
µ k (A) = sup<br />
ψ 1 ,...,ψ k ∈H<br />
<strong>in</strong>f {〈φ, Aφ〉 ; φ ∈ D(A), ‖φ‖ = 1, φ ⊥ ψ 1 , . . . , ψ k }<br />
(3.42)<br />
The opera<strong>to</strong>r A is bounded below iff µ 0 (A) > −∞ and µ 0 (A) is the <strong>in</strong>fimum of<br />
the spectrum of A.<br />
If A is bounded below and has purely discrete spectrum (i.e. σ ess (A) = ∅), we can<br />
order the eigenvalues of A <strong>in</strong> <strong>in</strong>creas<strong>in</strong>g order and repeat them accord<strong>in</strong>g <strong>to</strong> their<br />
multiplicity, namely<br />
E 0 (A) ≤ E 1 (A) ≤ E 2 (A) ≤ . . . . (3.43)<br />
If an eigenvalue E of A has multiplicity m it occurs <strong>in</strong> (3.43) exactly m times.<br />
The m<strong>in</strong>-max pr<strong>in</strong>ciple relates the E k (A) with the µ k (A).
THEOREM 3.1 (M<strong>in</strong>-max pr<strong>in</strong>ciple). If the self adjo<strong>in</strong>t opera<strong>to</strong>r A has purely discrete<br />
spectrum and is bounded below, then<br />
E k (A) = µ k (A) for all k ≥ 0 . (3.44)<br />
A proof of this important result can be found <strong>in</strong> [115]. The formulation there<br />
conta<strong>in</strong>s various ref<strong>in</strong>ements of our version. In particular [115] deals also with<br />
discrete spectrum below the <strong>in</strong>fimum of the essential spectrum.<br />
We state an application of Theorem 3.1. By A ≤ B we mean that the doma<strong>in</strong><br />
D(B) is a subset of the doma<strong>in</strong> D(A) and 〈φ, Aφ〉 ≤ 〈φ, Bφ〉 for all φ ∈ D(B).<br />
COROLLARY 3.2. Let A and B are self adjo<strong>in</strong>t opera<strong>to</strong>rs which are bounded below<br />
and have purely discrete spectrum. If A ≤ B then E k (A) ≤ E k (B) for all k.<br />
The Corollary follows directly from Theorem 3.1.<br />
We end this section with a short discussion of the S<strong>to</strong>ne-Weierstraß Theorem <strong>in</strong> the<br />
context of spectral theory. The S<strong>to</strong>ne-Weierstraß Theorem deals with subsets of<br />
the space C ∞ (R), the set of all (complex valued) cont<strong>in</strong>uous functions on R which<br />
vanish at <strong>in</strong>f<strong>in</strong>ity..<br />
A subset D is called an <strong>in</strong>volutative subalgebra of C ∞ (R), if it is a l<strong>in</strong>ear subspace<br />
and if for f, g ∈ D both the product f · g and the complex conjugate f belong <strong>to</strong><br />
D. We say that D seperates po<strong>in</strong>ts if for x, y ∈ R there is a function f ∈ D such<br />
that f(x) ≠ f(y) and both f(x) and f(y) are non zero.<br />
THEOREM 3.3 (S<strong>to</strong>ne-Weierstraß). If D is an <strong>in</strong>volutative subalgebra of C ∞ (R)<br />
which seperates po<strong>in</strong>ts, then D is dense <strong>in</strong> C ∞ (R) with respect <strong>to</strong> the <strong>to</strong>pology of<br />
uniform convergence.<br />
A proof of this theorem is conta<strong>in</strong>ed e.g. <strong>in</strong> [117]. Theorem 3.3 can be used <strong>to</strong><br />
prove some assertion P(f) for the opera<strong>to</strong>rs f(A) for all f ∈ C ∞ (R) if we know<br />
P(f) for some f. Suppose we know P(f) for all f ∈ D 0 . If we can show that<br />
D 0 seperates po<strong>in</strong>ts and that the set of all f satisfy<strong>in</strong>g P(f) is a closed <strong>in</strong>volutative<br />
subalgebra of C ∞ (R), then the S<strong>to</strong>ne-Weierstraß theorem tells us that P(f) holds<br />
for all f ∈ C ∞ (R).<br />
Theorem 3.3 is especially useful <strong>in</strong> connection with resolvents. Suppose a property<br />
P(f) holds for all functions f <strong>in</strong> R, the set of l<strong>in</strong>ear comb<strong>in</strong>ations of the functions<br />
f ζ (x) = 1<br />
x−ζ<br />
for all ζ ∈ C\R, so for resolvents of A and their l<strong>in</strong>ear comb<strong>in</strong>ations.<br />
The resolvent equations (or rather basic algebra of R) tell us that R is actually an<br />
<strong>in</strong>volutative algebra. So, if the property P(f) survives uniform limits, we can<br />
conclude that P(f) is valid for all f ∈ C ∞ (R). The above procedure was dubbed<br />
the ‘S<strong>to</strong>ne-Weierstraß Gavotte’ <strong>in</strong> [30]. More details can be found there.<br />
3.4. <strong>Random</strong> potentials.<br />
DEFINITION 3.4. A random variable is a real valued measurable function on a<br />
probability space (Ω, F, P).<br />
19
20<br />
If X is a random variable we call the probability measure P 0 on R def<strong>in</strong>ed by<br />
P 0 (A) = P ( {ω | X(ω) ∈ A } ) for any Borel set A (3.45)<br />
the distribution of X. If the distributions of the random variables X and Y agree<br />
we say that X and Y are identically distributed. We also say that X and Y have a<br />
common distribution <strong>in</strong> this case.<br />
A family {X i } i∈I of random variables is called <strong>in</strong>dependent if for any f<strong>in</strong>ite subset<br />
{i 1 , . . . , i n } of I<br />
( {ω|<br />
P Xi1 (ω) ∈ [a 1 , b 1 ], X i2 (ω) ∈ [a 2 , b 2 ], . . . X <strong>in</strong> (ω) ∈ [a n , b n ] })<br />
= P<br />
( {ω|<br />
Xi1 (ω) ∈ [a 1 , b 1 ] }) · . . . · P<br />
( {ω|<br />
X<strong>in</strong> (ω) ∈ [a n , b n ] }) . (3.46)<br />
REMARK 3.5. If X i are <strong>in</strong>dependent and identically distributed (iid) with common<br />
distribution P 0 then<br />
( {ω|<br />
P Xi1 (ω) ∈ [a 1 , b 1 ], X i2 (ω) ∈ [a 2 , b 2 ], . . . X <strong>in</strong> (ω) ∈ [a n , b n ] })<br />
= P 0 ([a 1 , b 1 ]) · P 0 ([a 2 , b 2 ]) · . . . · P 0 ([a n , b n ]) .<br />
For the reader’s convenience we state a very useful result of elementary probability<br />
theory which we will need a number of times <strong>in</strong> this text.<br />
THEOREM 3.6 (Borel-Cantelli lemma). Let (Ω, F, P) be a probability space and<br />
{A n } n∈N be a sequence of set <strong>in</strong> F. Denote by A ∞ the set<br />
(1) If<br />
A ∞ = {ω ∈ Ω | ω ∈ A n for <strong>in</strong>f<strong>in</strong>itely many n} (3.47)<br />
∑ ∞<br />
n=1 P(A n) < ∞, then P(A ∞ ) = 0<br />
(2) If the sets {A n } are <strong>in</strong>dependent<br />
and ∑ ∞<br />
n=1 P(A n) = ∞, then P(A ∞ ) = 1<br />
REMARK 3.7.<br />
(1) We recall that a sequence {A n } of events (i.e. of sets from F) is called<br />
<strong>in</strong>dependent if for any f<strong>in</strong>ite subsequence {A nj } j=1,...,M<br />
P ( M ⋂<br />
j=1<br />
A nj<br />
)<br />
=<br />
M ∏<br />
j=1<br />
(2) The set A ∞ can be written as A ∞ = ⋂ N<br />
For the proof of Theorem 3.6 see e.g. [12] or [95].<br />
P (A nj ) (3.48)<br />
⋃<br />
n≥N A n.<br />
From now on we assume that the random variables {V ω (n)} n∈Z d are <strong>in</strong>dependent<br />
and identically distributed with common distribution P 0 .<br />
By supp P 0 we denote the support of the measure P 0 , i.e.<br />
supp P 0 = {x ∈ R | P 0<br />
(<br />
(x − ε, x + ε)<br />
)<br />
> 0 for all ε > 0} . (3.49)<br />
If supp P 0 is compact then the opera<strong>to</strong>r H ω = H 0 + V ω is bounded. In fact, if<br />
supp P 0 ⊂ [−M, M], with probability one
21<br />
sup |V ω (j)| ≤ M .<br />
j∈Z d<br />
Even if supp P 0 is not compact the multiplication opera<strong>to</strong>r V ω is selfadjo<strong>in</strong>t on<br />
D = {ϕ ∈ l 2 |V ω ϕ ∈ l 2 }. It is essentially selfadjo<strong>in</strong>t on<br />
l 2 0(Z d ) = {ϕ ∈ l 2 (Z d ) | ϕ(i) = 0 for all but f<strong>in</strong>itely many po<strong>in</strong>ts i} .<br />
S<strong>in</strong>ce H 0 is bounded it is a fortiori Ka<strong>to</strong> bounded with respect <strong>to</strong> V ω . By the Ka<strong>to</strong>-<br />
Rellich theorem it follows that H ω = H 0 + V ω is essentially selfadjo<strong>in</strong>t on l 2 0 (Zd )<br />
as well (see [114] for details).<br />
In a first result about the <strong>An</strong>derson model we are now go<strong>in</strong>g <strong>to</strong> determ<strong>in</strong>e its spectrum<br />
(as a set). In particular we will see that the spectrum σ(H ω ) is (P-almost<br />
surely) a fixed non random set. First we prove a proposition which while easy is<br />
very useful <strong>in</strong> the follow<strong>in</strong>g. Roughly speak<strong>in</strong>g, this proposition tells us:<br />
Whatever can happen, will happen, <strong>in</strong> fact <strong>in</strong>f<strong>in</strong>itely often.<br />
PROPOSITION 3.8. There is a set Ω 0 of probability one such that the follow<strong>in</strong>g is<br />
true: For any ω ∈ Ω 0 , any f<strong>in</strong>ite set Λ ⊂ Z d , any sequence {q i } i∈Λ , q i ∈ supp P 0<br />
and any ε > 0, there exists a sequence {j n } <strong>in</strong> Z d with || j n || ∞ → ∞ such that<br />
sup | q i − V ω (i + j n ) | < ε .<br />
i∈Λ<br />
PROOF: Fix a f<strong>in</strong>ite set Λ, a sequence {q i } i∈Λ , q i ∈ supp P 0 and ε > 0.<br />
Then, by the def<strong>in</strong>ition of supp and the <strong>in</strong>dependence of the q i we have for A =<br />
{ω| sup i∈Λ |V ω (i) − q i | < ε}<br />
P(A) > 0 .<br />
Pick a sequence l n ∈ Z d , such that the distance between any l n , l m (n ≠ m) is<br />
bigger than twice the diameter of Λ. Then, the events<br />
A n = A n (Λ, {q i } i∈Λ , ε) = {ω| sup |V ω (i + l n ) − q i | < ε}<br />
i∈Λ<br />
are <strong>in</strong>dependent and P(A n ) = P(A) > 0. Consequently, the Borel-Cantelli lemma<br />
(see Theorem 3.6) tells us that<br />
Ω Λ,{qi },ε = { ω | ω ∈ A n for <strong>in</strong>f<strong>in</strong>itely many n}<br />
has probability one.<br />
The set supp P 0 conta<strong>in</strong>s a countable dense set R 0 . Moreover, the system Ξ of all<br />
f<strong>in</strong>ite subsets of Z d is countable. Thus the set<br />
Ω 0 :=<br />
⋂<br />
Λ ∈ Ξ,<br />
{q i }∈R 0 ,n∈N<br />
Ω Λ,{qi }, 1 n
22<br />
has probability one. It is a countable <strong>in</strong>tersection of sets of probability one.<br />
By its def<strong>in</strong>ition, Ω 0 satisfies the requirements of the assertion.<br />
□<br />
We now turn <strong>to</strong> the announced theorem<br />
THEOREM 3.9. For P-almost all ω we have σ(H ω ) = [0, 4d ] + supp P 0 .<br />
PROOF: The spectrum σ(V ) of the multiplication opera<strong>to</strong>r with V (n) is given<br />
by the closure of the set R(V ) = {V (n)|n ∈ Z d }. Hence σ(V ω ) = supp P 0 almost<br />
surely. S<strong>in</strong>ce 0 ≤ H 0 ≤ 4d we have<br />
σ(H 0 + V ω ) ⊂ σ(V ω ) + [0, ‖H 0 ‖]<br />
= supp P 0 + [0, 4d] .<br />
Let us prove the converse. We use the Weyl criterion (see [117] or [141]):<br />
λ ∈ σ(H ω ) ⇔ ∃ ϕ n ∈ D 0 , ||ϕ n || = 1 : ||(H ω − λ)ϕ n || → 0 ,<br />
where D 0 is any vec<strong>to</strong>r space such that H ω is essentially selfadjo<strong>in</strong>t on D 0 . The<br />
sequence ϕ n is called a Weyl sequence. In a sense, ϕ n is an ‘approximate eigenfunction’.<br />
Let λ ∈ [0, 4d] + supp P 0 , say λ = λ 0 + λ 1 , λ 0 ∈ σ(H 0 ) = [0, 4d], λ 1 ∈ supp P 0 .<br />
Take a Weyl sequence ϕ n for H 0 and λ 0 , i. e. ||(H 0 − λ 0 )ϕ n || → 0, ||ϕ n || = 1.<br />
S<strong>in</strong>ce H 0 is essentially selfadjo<strong>in</strong>t on D 0 = l 2 0 (Zd ) (<strong>in</strong> fact H 0 is bounded), we<br />
may suppose ϕ n ∈ D 0 . Sett<strong>in</strong>g ϕ (j) (i) = ϕ(i − j), we easily see<br />
H 0 ϕ (j) = (H 0 ϕ) (j) .<br />
Due <strong>to</strong> Proposition 3.8 there is (with probability one) a sequence {j n }, || j n || ∞ → ∞<br />
such that<br />
sup<br />
i∈supp ϕ n<br />
|V ω (i + j n ) − λ 1 | < 1 n . (3.50)<br />
Def<strong>in</strong>e ψ n = ϕ jn<br />
n . Then ψ n is a Weyl sequence for H ω and λ = λ 0 + λ 1 . This<br />
proves the theorem.<br />
□<br />
The above result tells us <strong>in</strong> particular that the spectrum σ(H ω ) is (almost surely)<br />
non random. Moreover, an <strong>in</strong>spection of the proof shows that there is no discrete<br />
spectrum (almost surely), as the constructed Weyl sequence tends <strong>to</strong> zero weakly,<br />
<strong>in</strong> fact can be chosen <strong>to</strong> be orthonormal. Both results are valid <strong>in</strong> much bigger<br />
generality. They are due <strong>to</strong> ergodicity properties of the potential V ω . We will<br />
discuss this <strong>to</strong>pic <strong>in</strong> the follow<strong>in</strong>g chapter.<br />
Notes and Remarks<br />
For further <strong>in</strong>formation see [23] and [30] or consult [94] and [64, 65].
23<br />
4.1. Ergodic s<strong>to</strong>chastic processes.<br />
4. Ergodicity properties<br />
Some of the basic questions about random Schröd<strong>in</strong>ger can be formulated and answered<br />
most conveniently with<strong>in</strong> the framework of ‘ergodic opera<strong>to</strong>rs’. This class<br />
of opera<strong>to</strong>rs comprises many random opera<strong>to</strong>rs, such as the <strong>An</strong>derson model and<br />
its cont<strong>in</strong>uous analogs, the Poisson model, as well as random acoustic opera<strong>to</strong>rs.<br />
Moreover, also opera<strong>to</strong>rs with almost periodic potentials can be viewed as ergodic<br />
opera<strong>to</strong>rs.<br />
In these notes we only briefly <strong>to</strong>uch the <strong>to</strong>pic of ergodic opera<strong>to</strong>rs. We just collect a<br />
few def<strong>in</strong>itions and results we will need <strong>in</strong> the follow<strong>in</strong>g chapters. We recommend<br />
the references cited <strong>in</strong> the notes at the end of this chapter for further read<strong>in</strong>g.<br />
Ergodic s<strong>to</strong>chastic processes are a certa<strong>in</strong> generalization of <strong>in</strong>dependent, identically<br />
distributed random variables. The assumption that the random variables X i<br />
and X j are <strong>in</strong>dependent for |i − j| > 0 is replaced by the requirement that X i and<br />
X j are ‘almost <strong>in</strong>dependent’ if |i − j| is large (see the discussion below, especially<br />
(4.2), for a precise statement). The most important result about ergodic processes<br />
is the ergodic theorem (see Theorem 4.2 below), which says that the strong law of<br />
large numbers, one of the basic results about <strong>in</strong>dependent, identically distributed<br />
random variables, extends <strong>to</strong> ergodic processes.<br />
At a number a places <strong>in</strong> these notes we will have <strong>to</strong> deal with ergodic processes.<br />
Certa<strong>in</strong> important quantities connected with random opera<strong>to</strong>rs are ergodic but not<br />
<strong>in</strong>dependent even if the potential V ω is a sequence of <strong>in</strong>dependent random variables.<br />
A family {X i } i∈Z d of random variables is called a s<strong>to</strong>chastic process (with <strong>in</strong>dex<br />
set Z d ). This means that there is a probability space (Ω, F, P) (F a σ-algebra<br />
on Ω and P a probability measure on (Ω, F)) such that the X i are real valued,<br />
measurable functions on (Ω, F).<br />
The quantities of <strong>in</strong>terest are the probabilities of events that can be expressed<br />
through the random variables X i , like<br />
1<br />
{ω| lim<br />
N→∞ |Λ N |<br />
∑<br />
|| i|| ∞≤N<br />
X i (ω) = 0} .<br />
The special way Ω is constructed is irrelevant. For example, one may take the set<br />
R Zd as Ω. The correspond<strong>in</strong>g σ-algebra F is generated by cyl<strong>in</strong>der sets of the<br />
form<br />
{ω | ω i1 ∈ A 1 , . . . , ω <strong>in</strong> ∈ A n } (4.1)<br />
where A 1 , . . . , A n are Borel subsets of R. On Ω the random variables X i can be<br />
realized by X i (ω) = ω i .<br />
This choice of (Ω, F) is called the canonical probability space . For details <strong>in</strong><br />
connection with random opera<strong>to</strong>rs see e.g. [58, 64]. Given a probability space<br />
(Ω, F, P) we call a measurable mapp<strong>in</strong>g T : Ω → Ω a measure preserv<strong>in</strong>g transformation<br />
if P(T −1 A) = P(A) for all A ∈ F. If {T i } i∈Z d is a family of
24<br />
measure preserv<strong>in</strong>g transformations we call a set A ∈ F <strong>in</strong>variant (under {T i }) if<br />
Ti −1 A = A for all i ∈ Z d .<br />
A family {T i } of measure preserv<strong>in</strong>g transformations on a probability space (Ω, F, P)<br />
is called ergodic (with respect <strong>to</strong> the probability measure P) if any <strong>in</strong>variant A ∈ F<br />
has probability zero or one. A s<strong>to</strong>chastic process {X i } i∈Z d is called ergodic, if<br />
there exists an ergodic family of measure preserv<strong>in</strong>g transformations {T i } i∈Z d such<br />
that X i (T j ω) = X i−j (ω).<br />
Our ma<strong>in</strong> example of an ergodic s<strong>to</strong>chastic process is given by <strong>in</strong>dependent, identically<br />
distributed random variables X i (ω) = V ω (i) (a random potential on Z d ).<br />
Due <strong>to</strong> the <strong>in</strong>dependence of the random variables the probability measure P (on<br />
Ω = R Zd ) is just the <strong>in</strong>f<strong>in</strong>ite product measure of the probability measure P 0 on R<br />
given by P 0 (M) = P(V ω (0) ∈ M). P 0 is the distribution of V ω (0).<br />
It is easy <strong>to</strong> see that the shift opera<strong>to</strong>rs<br />
(T i ω) j = ω j−i<br />
form a family of measure preserv<strong>in</strong>g transformations on R Zd <strong>in</strong> this case.<br />
It is not hard <strong>to</strong> see that the family of shift opera<strong>to</strong>rs is ergodic with respect <strong>to</strong> the<br />
product measure P. One way <strong>to</strong> prove this is <strong>to</strong> show that<br />
P(T −1<br />
i<br />
A ∩ B) → P(A) P(B) (4.2)<br />
as || i|| ∞ → ∞ for all A, B ∈ F. This is obvious if both A and B are of the form<br />
(4.1). Moreover, the system of sets A, B for which (4.2) holds is a σ-algebra, thus<br />
(4.2) is true for the σ-algebra generated by sets of the form (4.1), i.e. on F.<br />
Now let M be an <strong>in</strong>variant set. Then (4.2) (with A = B = M) gives<br />
P(M) = P(M ∩ M) = P(T −1<br />
i<br />
M ∩ M) → P(M) 2<br />
prov<strong>in</strong>g that M has probability zero or one.<br />
We will need two more results on ergodicity.<br />
PROPOSITION 4.1. Let {T i } i∈Z d be an ergodic family of measure preserv<strong>in</strong>g transformations<br />
on a probability space (Ω, F, P ). If a random variable Y is <strong>in</strong>variant<br />
under {T i } (i.e. Y (T i ω) = Y (ω) for all i ∈ Z d ) then Y is almost surely constant,<br />
i.e. there is a c ∈ R, such that P(Y = c) = 1.<br />
We may allow the values ±∞ for Y (and hence for c) <strong>in</strong> the above result.The proof<br />
is not difficult (see e.g. [30]).<br />
The f<strong>in</strong>al result is the celebrated ergodic theorem by Birkhoff. It generalizes the<br />
strong law of large number <strong>to</strong> ergodic processes.<br />
We denote by E(·) the expectation with respect <strong>to</strong> the probability measure P.<br />
THEOREM 4.2. If {X i } i∈Z d is an ergodic process and E(|X 0 |) < ∞ then<br />
1 ∑<br />
lim<br />
L→∞ (2L + 1) d X i → E(X 0 )<br />
i∈Λ L<br />
for P-almost all ω.
For a proof of this fundamental result see e.g. [96]. We remark that the ergodic<br />
theorem has important extensions <strong>in</strong> various directions (see [91]).<br />
25<br />
4.2. Ergodic opera<strong>to</strong>rs.<br />
Let V ω (n), n ∈ Z d be an ergodic process (for example, one may th<strong>in</strong>k of <strong>in</strong>dependent<br />
identically distributed V ω (n)).<br />
Then there exist measure preserv<strong>in</strong>g transformations {T i } on Ω such that<br />
(1) V ω (n) satisfies<br />
V Ti ω(n) = V ω (n − i) . (4.3)<br />
(2) <strong>An</strong>y measurable subset of Ω which is <strong>in</strong>variant under the {T i } has trivial<br />
probability (i.e. P(A) = 0 or P(A) = 1) .<br />
We def<strong>in</strong>e translation opera<strong>to</strong>rs {U i } i∈Z d on l 2 (Z d ) by<br />
(U i ϕ) m = ϕ m−i , ϕ ∈ l 2 (Z d ) . (4.4)<br />
It is clear that the opera<strong>to</strong>rs U i are unitary. Moreover, if we denote the multiplication<br />
opera<strong>to</strong>rs with the function V by V then<br />
V Ti ω = U i V ω U ∗ i . (4.5)<br />
The free Hamil<strong>to</strong>nian H 0 of the <strong>An</strong>derson model (3.7) commutes with U i , thus<br />
(4.5) implies<br />
H Ti ω = U i H ω U ∗ i . (4.6)<br />
i.e. H Ti ω and H ω are unitarily equivalent.<br />
Opera<strong>to</strong>rs satisfy<strong>in</strong>g (4.6) (with ergodic T i and unitary U i ) are called ergodic opera<strong>to</strong>rs<br />
.<br />
The follow<strong>in</strong>g result is basic <strong>to</strong> the theory of ergodic opera<strong>to</strong>rs.<br />
THEOREM 4.3. (Pastur) If H ω is an ergodic family of selfadjo<strong>in</strong>t opera<strong>to</strong>rs, then<br />
there is a (closed, nonrandom) subset Σ of R, such that<br />
σ(H ω ) = Σ for P-almost all ω.<br />
Moreover, there are sets Σ ac , Σ sc , Σ pp such that<br />
REMARK 4.4.<br />
σ ac (H ω ) = Σ ac , σ sc (H ω ) = Σ sc , σ pp (H ω ) = Σ pp<br />
for P-almost all ω.<br />
(1) The theorem <strong>in</strong> its orig<strong>in</strong>al form is due <strong>to</strong> Pastur [111]. It was extended<br />
<strong>in</strong> [94] and [64].<br />
(2) We have been sloppy about the measurability properties of H ω which<br />
have <strong>to</strong> be def<strong>in</strong>ed and checked carefully. They are satisfied <strong>in</strong> our case<br />
(i.e. for the <strong>An</strong>derson model). For a precise formulation and proofs see<br />
[64].
26<br />
(3) We denote by σ ac (H), σ sc (H), σ pp (H) the absolutely cont<strong>in</strong>uous (resp.<br />
s<strong>in</strong>gularly cont<strong>in</strong>uous, resp. pure po<strong>in</strong>t) spectrum of the opera<strong>to</strong>r H. For<br />
a def<strong>in</strong>ition and basic poperties we refer <strong>to</strong> Sections 7.2and 7.3.<br />
PROOF (Sketch) : If H ω is ergodic and f is a bounded (measurable) function<br />
then f(H ω ) is ergodic as well, i.e.<br />
f(H Ti w) = U i f(H ω )U ∗ i .<br />
(see Lemma 4.5).<br />
We have (λ, µ) ∩ σ(H ω ) ≠ ∅ if and only if χ (λ,µ) (H ω ) ≠ 0.<br />
This is equivalent <strong>to</strong> Y λ,µ (ω) := tr χ (λ,µ) (H ω ) ≠ 0.<br />
S<strong>in</strong>ce χ (λ,µ) (H ω ) is ergodic, Y λ,µ is an <strong>in</strong>variant random variable and consequently,<br />
by Proposition 4.1 Y λ,µ = c λ,µ for all ω ∈ Ω λ,µ with P (Ω λ,µ ) = 1.<br />
Set<br />
Ω 0 =<br />
⋂<br />
Ω λ,µ .<br />
λ,µ∈Q, λ≤µ<br />
S<strong>in</strong>ce Ω 0 is a countable <strong>in</strong>tersection of sets of full measure, it follows that P (Ω 0 ) =<br />
1. Hence we can set<br />
Σ = {E | c λ,µ ≠ 0 for all λ < E < µ, λ, µ ∈ Q} .<br />
To prove the assertions on σ ac we need that the projection on<strong>to</strong> H ac , the absolutely<br />
cont<strong>in</strong>uous subspace with respect <strong>to</strong> H ω is measurable, the rest is as above. The<br />
same is true for σ sc and σ pp .<br />
We omit the measurability proof and refer <strong>to</strong> [64] or [23].<br />
□<br />
Above we used the follow<strong>in</strong>g results<br />
LEMMA 4.5. Let A be a self adjo<strong>in</strong>t opera<strong>to</strong>rs and U a unitary opera<strong>to</strong>r, then for<br />
any bounded measurable function f we have<br />
f ( UAU ∗) = U f(A) U ∗ . (4.7)<br />
PROOF: For resolvents, i.e. for f z (λ) = 1<br />
λ−z<br />
with z ∈ C \ R equation (4.7)<br />
can be checked directly. L<strong>in</strong>ear comb<strong>in</strong>ations of the f z are dense <strong>in</strong> C ∞ (R), the<br />
cont<strong>in</strong>uous functions vanish<strong>in</strong>g at <strong>in</strong>f<strong>in</strong>ity, by the S<strong>to</strong>ne-Weierstraß theorem (see<br />
Section 3.3). Thus (4.7) is true for f ∈ C ∞ (R).<br />
If µ and ν are the projection valued measures for A and B = UAU ∗ respectively,<br />
we have therefore for all f ∈ C ∞ (R)<br />
∫<br />
f(λ) d ν ϕ,ψ (λ) = 〈ϕ, f(B) ψ〉<br />
= 〈ϕ, Uf(A) U ∗ ψ〉<br />
= 〈U ∗ ϕ, f(A) U ∗ ψ〉<br />
∫<br />
= f(λ) d µ U ∗ ϕ,U ∗ ψ(λ) (4.8)
holds for all . Thus the measures µ ϕ,ψ and ν U ∗ ϕ,U ∗ ψ agree. Therefore (4.8) holds<br />
for all bounded measurable f.<br />
□<br />
27<br />
Notes and Remarks<br />
For further <strong>in</strong>formation see [23], [58], [64], [65], [94], [111] and [112]. <strong>An</strong> recent<br />
extensive review on ergodic opera<strong>to</strong>rs can be found <strong>in</strong> [55].
29<br />
5.1. Def<strong>in</strong>ition and existence.<br />
5. The density of states<br />
Here, as <strong>in</strong> the rest of the paper we consider the <strong>An</strong>derson model, i.e.<br />
H ω = H 0 + V ω on l 2 (Z d ) with <strong>in</strong>dependent random variables V ω (n) with a common<br />
distribution P 0 .<br />
In this section we def<strong>in</strong>e a quantity of fundamental importance for models <strong>in</strong><br />
condensed matter physics: the density of states. The density of states measure<br />
ν([E 1 , E 2 ]) gives the ‘number of states per unit volume’ with energy between E 1<br />
and E 2 . S<strong>in</strong>ce the spectrum of our Hamil<strong>to</strong>nian H ω is not discrete we can not<br />
simply count eigenvalues with<strong>in</strong> the <strong>in</strong>terval [E 1 , E 2 ] or, what is the same, take the<br />
dimension of the correspond<strong>in</strong>g spectral projection. In fact, the dimension of any<br />
spectral projection of H ω is either zero or <strong>in</strong>f<strong>in</strong>ite. Instead we restrict the spectral<br />
projection <strong>to</strong> the f<strong>in</strong>ite cube Λ L (see 3.5) <strong>in</strong> Z d , take the dimension of its range<br />
and divide by | Λ L | = (2L + 1) d the number of po<strong>in</strong>ts <strong>in</strong> Λ L . F<strong>in</strong>ally, we send<br />
the parameter L <strong>to</strong> <strong>in</strong>f<strong>in</strong>ity. This procedure is sometimes called the thermodynamic<br />
limit.<br />
For any bounded measurable function ϕ on the real l<strong>in</strong>e we def<strong>in</strong>e the quantity<br />
ν L (ϕ) = 1<br />
|Λ L | tr ( χ Λ L<br />
ϕ(H ω ) χ ΛL ) = 1<br />
|Λ L | tr ( ϕ(H ω) χ ΛL ) . (5.1)<br />
Here χ Λ denotes the characteristic function of the set Λ, (i.e. χ Λ (x) = 1 for<br />
x ∈ Λ and = 0 otherwise). The opera<strong>to</strong>rs ϕ(H ω ) are def<strong>in</strong>ed via the spectral<br />
theorem (see Section 3.2). In equation (5.1) we used the cyclicity of the trace, (i.e.:<br />
tr(AB) = tr(BA)) and the fact that χ 2 Λ = χ Λ .<br />
S<strong>in</strong>ce ν L is a positive l<strong>in</strong>ear functional on the bounded cont<strong>in</strong>uous functions, by<br />
Riesz representation theorem, it comes from a measure which we also call ν L , i.e.<br />
∫<br />
ν L (ϕ) = ϕ(λ) dν L (λ). (5.2)<br />
R<br />
We will show <strong>in</strong> the follow<strong>in</strong>g that the measures ν L converge <strong>to</strong> a limit measure ν<br />
as L → ∞ <strong>in</strong> the sense of vague convergence of measures for P-almost all ω.<br />
DEFINITION 5.1. A series ν n of Borel measures on R is said <strong>to</strong> converge vaguely<br />
<strong>to</strong> a Borel measure ν if ∫<br />
∫<br />
ϕ(x) d ν n (x) →<br />
ϕ(x) d ν(x)<br />
for all function ϕ ∈ C 0 (R), the set of cont<strong>in</strong>uous functions with compact support.<br />
We start with a proposition which establishes the almost sure convergence of the<br />
<strong>in</strong>tegral of ν L over a given function.<br />
PROPOSITION 5.2. If ϕ is a bounded measurable function, then for P-almost all ω<br />
lim<br />
L→∞<br />
1<br />
|Λ L | tr ( ϕ(H ω ) χ ΛL<br />
)<br />
= E<br />
(<br />
〈δ0 , ϕ(H ω )δ 0 〉 ) . (5.3)
30<br />
REMARK 5.3. The right hand side of (5.3) def<strong>in</strong>es a positive measure ν by<br />
∫<br />
ϕ(λ) dν(λ) = E(〈δ 0 , ϕ(H ω )δ 0 〉) .<br />
This measure satisfies ν(R) = 1, hence it is a probability measure (just <strong>in</strong>sert<br />
ϕ(λ) ≡ 1).<br />
DEFINITION 5.4. The measure ν, def<strong>in</strong>ed by<br />
ν(A) = E ( 〈δ 0 , χ A (H ω ) δ 0 〉 ) for A a Borel set <strong>in</strong> R (5.4)<br />
is called the density of states measure .<br />
The distribution function N of ν, def<strong>in</strong>ed by<br />
is known as the <strong>in</strong>tegrated density of states .<br />
PROOF (Proposition) :<br />
=<br />
N(E) = ν ( (−∞, E] ) (5.5)<br />
1<br />
|Λ L | tr (ϕ(H ω)χ ΛL )<br />
1 ∑<br />
(2L + 1) d<br />
i∈Λ L<br />
〈δ i , ϕ(H ω )δ i 〉 (5.6)<br />
The random variables X i = 〈δ i , ϕ(H ω )δ i 〉 form an ergodic s<strong>to</strong>chastic process s<strong>in</strong>ce<br />
the shift opera<strong>to</strong>rs {T i } are ergodic and s<strong>in</strong>ce<br />
X i (T j ω) = 〈δ i , ϕ(H Tj ω) δ i 〉<br />
= 〈δ i , U j ϕ(H ω ) U ∗ j δ i 〉<br />
= 〈U ∗ j δ i , ϕ(H ω ) U ∗ j δ i 〉<br />
= 〈δ i−j , ϕ(H ω ) δ i−j 〉<br />
= X i−j (ω) . (5.7)<br />
We used that U ∗ j δ i(n) = δ i (n + j) = δ i−j (n).<br />
S<strong>in</strong>ce |X i | ≤ ||ϕ|| ∞ , the X i are <strong>in</strong>tegrable (with respect <strong>to</strong> P). Thus we may apply<br />
the ergodic theorem (4.2) <strong>to</strong> obta<strong>in</strong><br />
1<br />
|Λ L | tr(ϕ(H ω)χ ΛL ) =<br />
1 ∑<br />
(2L + 1) d<br />
i∈Λ L<br />
X i (5.8)<br />
−→ E(X 0 ) = E(〈δ 0 , ϕ(H ω )δ 0 〉) . (5.9)<br />
We have proven that (5.3) holds for fixed ϕ on a set of full probability. This set,<br />
let’s call it Ω ϕ , may (and will) depend on ϕ. We can conclude that (5.3) holds<br />
for all ϕ for ω ∈ ⋂ ϕ Ω ϕ. However, this is an uncountable <strong>in</strong>tersection of sets of<br />
□
probability one. We do not know whether this <strong>in</strong>tersection has full measure, <strong>in</strong> fact<br />
we even don’t know whether this set is measurable.<br />
THEOREM 5.5. The measures ν L converge vaguely <strong>to</strong> the measure ν P-almost<br />
surely, i.e. there is a set Ω 0 of probability one, such that<br />
∫<br />
∫<br />
ϕ(λ) dν L (λ) → ϕ(λ) dν(λ) (5.10)<br />
for all ϕ ∈ C 0 (R) and all ω ∈ Ω 0 .<br />
REMARK 5.6. The measure ν is non random by def<strong>in</strong>ition.<br />
PROOF: Take a countable dense set D 0 <strong>in</strong> C 0 (R) <strong>in</strong> the uniform <strong>to</strong>pology. With<br />
Ω ϕ be<strong>in</strong>g the set of full measure for which (5.10) holds, we set<br />
Ω 0 = ⋂<br />
Ω ϕ .<br />
ϕ∈D 0<br />
S<strong>in</strong>ce Ω 0 is a countable <strong>in</strong>tersection of sets of full measure, Ω 0 has probability one.<br />
For ω ∈ Ω 0 the convergence (5.10) holds for all ϕ ∈ D 0 .<br />
By assumption on D 0 , if ϕ ∈ C 0 (R) there is a sequence ϕ n ∈ D 0 with ϕ n → ϕ<br />
uniformly. It follows<br />
∫<br />
|<br />
∫<br />
≤ |<br />
∫<br />
+ |<br />
∫<br />
+ |<br />
∫<br />
ϕ(λ) dν(λ) −<br />
∫<br />
ϕ(λ) dν L (λ)|<br />
ϕ(λ) dν(λ) − ϕ n (λ) dν(λ)|<br />
∫<br />
ϕ n (λ) dν(λ) − ϕ n (λ) dν L (λ)|<br />
∫<br />
ϕ n (λ) dν L (λ) − ϕ(λ) dν L (λ)|<br />
31<br />
≤<br />
|| ϕ − ϕ n || ∞ · ν(R) + || ϕ − ϕ n || ∞ · ν L (R)<br />
∫<br />
∫<br />
+ | ϕ n (λ) dν(λ) − ϕ n (λ) dν L (λ)| . (5.11)<br />
S<strong>in</strong>ce both ν(R) and ν L (R) are bounded by 1 (<strong>in</strong> fact are equal <strong>to</strong> one) the first two<br />
terms can be made small by tak<strong>in</strong>g n large enough. We make the third term small<br />
by tak<strong>in</strong>g L large.<br />
□<br />
REMARKS 5.7.<br />
(1) As we remarked already <strong>in</strong> the above proof both ν L and ν are probability<br />
measures. Consequently, the measures ν L converge even weakly <strong>to</strong> ν, i. e.<br />
when <strong>in</strong>tegrated aga<strong>in</strong>st a bounded cont<strong>in</strong>uous function (see e. g. [12]).<br />
Observe that the space of bounded cont<strong>in</strong>uous functions C b (R) does not<br />
conta<strong>in</strong> a countable dense set, so the above proof does not work for C b<br />
directly.
32<br />
(2) In the cont<strong>in</strong>uous case the density of states measure is unbounded, even<br />
for the free Hamil<strong>to</strong>nian. So, <strong>in</strong> the cont<strong>in</strong>uous case, it does not make<br />
sense even <strong>to</strong> talk about weak convergence, we have <strong>to</strong> restrict ourselves<br />
<strong>to</strong> vague convergence <strong>in</strong> this case.<br />
(3) Given a countable set D of bounded measurable functions we can f<strong>in</strong>d a<br />
set Ω 1 of probability one such that<br />
∫<br />
∫<br />
ϕ(λ)dν L (λ) → ϕ(λ)dν(λ)<br />
for all ϕ ∈ D ∪ C b (R) and all ω ∈ Ω 1 .<br />
COROLLARY 5.8. For P-almost all ω the follow<strong>in</strong>g is true:<br />
For all E ∈ R<br />
N(E) = lim ν ( )<br />
L (−∞, E] . (5.12)<br />
L→∞<br />
REMARKS 5.9. It is an immediate consequence of Proposition 5.2 that for fixed E<br />
the convergence <strong>in</strong> (5.12) holds for almost all ω, with the set of exceptional ω be<strong>in</strong>g<br />
E-dependent. The statement of Corollary 5.8 is stronger: It claims the existence of<br />
an E-<strong>in</strong>dependent set of ω such that 5.12 is true for all E.<br />
PROOF: We will prove (5.12) first for energies E where N is cont<strong>in</strong>uous.<br />
S<strong>in</strong>ce N is mono<strong>to</strong>ne <strong>in</strong>creas<strong>in</strong>g the set of discont<strong>in</strong>uity po<strong>in</strong>ts of N is at most<br />
countable (see Lemma 5.10 below). Consequently, there is a countable set S of<br />
cont<strong>in</strong>uity po<strong>in</strong>ts of N which is dense <strong>in</strong> R. By Proposition 5.2 there is a set of full<br />
P-measure such that<br />
∫<br />
χ (−∞,E] (λ) dν L (λ) → N(E)<br />
for all E ∈ S.<br />
Take ε > 0. Suppose E is an arbitrary cont<strong>in</strong>uity po<strong>in</strong>t of N. Then, we f<strong>in</strong>d<br />
E + , E − ∈ S with E − ≤ E ≤ E + such that N(E + ) − N(E − ) < ε 2 .<br />
We estimate (N is mono<strong>to</strong>ne <strong>in</strong>creas<strong>in</strong>g)<br />
∫<br />
N(E) − χ (−∞,E] (λ) dν L (λ) (5.13)<br />
∫<br />
≤ N(E + ) − χ (−∞,E− ](λ) dν L (λ) (5.14)<br />
≤ N(E + ) − N(E − ) + ∣ ∫<br />
N(E− ) − χ (−∞,E− ](λ) dν L (λ) ∣ (5.15)<br />
for L large enough.<br />
≤ ε (5.16)
<strong>An</strong>alogously we get<br />
∫<br />
N(E) − χ (−∞,E] (λ) dν L (λ) (5.17)<br />
≥ N(E − ) − N(E + ) − ∣ ∫<br />
∣N(E + ) − χ (−∞,E+ ](λ) dν L (λ) ∣ (5.18)<br />
≥ −ε . (5.19)<br />
33<br />
Hence<br />
∫<br />
∣ N(E) −<br />
χ (−∞,E] (λ) dν L (λ) ∣ ∣ → 0<br />
This proves (5.12) for cont<strong>in</strong>uity po<strong>in</strong>ts. S<strong>in</strong>ce there are at most countably many<br />
po<strong>in</strong>ts of discont<strong>in</strong>uity for N another application of Proposition 5.2 proves the<br />
result for all E.<br />
□<br />
Above we used the follow<strong>in</strong>g Lemma.<br />
LEMMA 5.10. If the function F : R → R is mono<strong>to</strong>ne <strong>in</strong>creas<strong>in</strong>g then F has at<br />
most countably many po<strong>in</strong>ts of discont<strong>in</strong>uity.<br />
PROOF: S<strong>in</strong>ce F is mono<strong>to</strong>ne both F (t−) = lim s↗t F (s) and F (t+) = lim s↘t F (s)<br />
exist. If F is discont<strong>in</strong>uous at t ∈ R then F (t+) − F (t−) > 0. Set<br />
D n = {t ∈ R | F (t+) − F (t−) > 1 n }<br />
then the set D of discont<strong>in</strong>uity po<strong>in</strong>ts of F is given by ⋃ n∈N D n.<br />
Let us assume that D is uncountable. Then also one of the D n must be uncountable.<br />
S<strong>in</strong>ce F is mono<strong>to</strong>ne and def<strong>in</strong>ed on all of R it must be bounded on any bounded<br />
<strong>in</strong>terval. Thus we conclude that D n ∩ [−M, M] is f<strong>in</strong>ite for any M. It follows<br />
that D n = ⋃ (<br />
M∈N Dn ∩ [−M, M] ) is countable. This is a contradiction <strong>to</strong> the<br />
conclusion above.<br />
□<br />
REMARK 5.11.<br />
The proof of Corollary 5.8 shows that we also have<br />
N(E−) = sup N(E − ε)<br />
=<br />
ε>0<br />
∫<br />
χ (−∞,E) (λ) dν(λ)<br />
∫<br />
= lim<br />
L→∞<br />
χ (−∞,E) (λ) dν L (λ) (5.20)<br />
for all E and P-almost all ω (with an E-<strong>in</strong>dependent set of ω).<br />
Consequently, we also have ν({E}) = lim L→∞ ν L ({E}).<br />
PROPOSITION 5.12. supp(ν) = Σ (= σ(H ω )).
34<br />
PROOF: If λ /∈ Σ then there is an ɛ > 0 such that χ (λ−ɛ,λ+ɛ) (H ω ) = 0<br />
P-almost surely, hence<br />
ν ( (λ − ɛ, λ + ɛ) ) = E ( χ (λ−ɛ,λ+ɛ) (H ω )(0, 0) ) = 0 .<br />
If λ ∈ Σ then χ (λ−ɛ,λ+ɛ) (H ω ) ≠ 0 P-almost surely for any ɛ > 0.<br />
S<strong>in</strong>ce χ (λ−ɛ,λ+ɛ) (H ω ) is a projection, it follows that for some j ∈ Z d<br />
0 ≠ E ( χ (λ−ɛ,λ+ɛ) (H ω )(j, j) )<br />
Here, we used that by Lemma 4.5<br />
= E ( χ (λ−ɛ,λ+ɛ) (H ω )(0, 0) )<br />
= ν ( (λ − ɛ, λ + ɛ) ) . (5.21)<br />
f(H ω )(j, j) = f(H Tj ω)(0, 0)<br />
and the assumption that T j is measure preserv<strong>in</strong>g.<br />
It is not hard <strong>to</strong> see that the <strong>in</strong>tegrated density of states N(λ) is a cont<strong>in</strong>uous function,<br />
which is equivalent <strong>to</strong> the assertion that ν has no a<strong>to</strong>ms, i.e. ν({λ}) = 0 for<br />
all λ. We note, that an analogous result for the cont<strong>in</strong>uous case (i.e. Schröd<strong>in</strong>ger<br />
opera<strong>to</strong>rs on L 2 (R d )) is unknown <strong>in</strong> this generality.<br />
We first state<br />
LEMMA 5.13. Let V λ be the eigenspace of H ω with respect <strong>to</strong> the eigenvalue λ<br />
then dim ( χ ΛL (V λ )) ≤ CL d−1 .<br />
From this we deduce<br />
THEOREM 5.14. For any λ ∈ R ν({λ}) = 0.<br />
PROOF (of the Theorem assum<strong>in</strong>g the Lemma) :<br />
By Proposition 5.2 and Theorem 5.5 we have<br />
1<br />
ν({λ}) = lim<br />
L→∞ (2L + 1) d tr ( χ Λ L<br />
χ {λ} (H ω )). (5.22)<br />
If f i is an orthonormal basis of χ ΛL (V λ ) and g j an orthonormal basis of χ ΛL (V λ ) ⊥<br />
we have, not<strong>in</strong>g that χ ΛL (V λ ) is f<strong>in</strong>ite dimensional,<br />
tr ( χ ΛL χ {λ} (H ω ) )<br />
= ∑ i<br />
〈f i , χ ΛL χ {λ} (H ω )f i 〉 + ∑ j<br />
〈g j , χ ΛL χ {λ} (H ω )g j 〉<br />
□<br />
= ∑ i<br />
〈f i , χ ΛL χ {λ} (H ω )f i 〉<br />
≤ dim χ ΛL (V λ ) ≤ CL d−1 (5.23)<br />
hence (5.22) converges <strong>to</strong> zero. Thus ν ( {λ} ) = 0.<br />
□
PROOF (Lemma) :<br />
We def<strong>in</strong>e ˜Λ L = {i ∈ Λ L | (L − 1) ≤ || i|| ∞ ≤ L }<br />
˜Λ L consists of the two outermost layers of Λ L .<br />
The values u(n) of an eigenfunction u of H ω with H ω u = λu can be computed<br />
from the eigenvalue equation for all n ∈ Λ L once we know its values on ˜Λ L . So,<br />
the dimension of χ ΛL (V λ ) is at most the number of po<strong>in</strong>ts <strong>in</strong> ˜Λ L .<br />
□<br />
35<br />
5.2. Boundary conditions.<br />
Boundary conditions are used <strong>to</strong> def<strong>in</strong>e differential opera<strong>to</strong>rs on sets M with a<br />
boundary. A rigorous treatment of boundary conditions for differential opera<strong>to</strong>rs is<br />
most conveniently based on a quadratic form approach (see [115]) and is out of the<br />
scope of this review. Roughly speak<strong>in</strong>g boundary conditions restrict the doma<strong>in</strong><br />
of a differential opera<strong>to</strong>r D by requir<strong>in</strong>g that functions <strong>in</strong> the doma<strong>in</strong> of D have<br />
a certa<strong>in</strong> behavior at the boundary of M. In particular, Dirichlet boundary conditions<br />
force the functions f <strong>in</strong> the doma<strong>in</strong> <strong>to</strong> vanish at ∂M. Neumann boundary<br />
conditions require the normal derivative <strong>to</strong> vanish at the boundary. Let us denote<br />
by −∆ D M and −∆N M<br />
the Laplacian on M with Dirichlet and Neumann boundary<br />
condition respectively.<br />
The def<strong>in</strong>ition of boundary conditions for the discrete case are somewhat easier<br />
then <strong>in</strong> the cont<strong>in</strong>uous case. However, they are presumably less familiar <strong>to</strong> the<br />
reader and may look somewhat technical at a first glance. The reader might therefore<br />
skip the details for the first read<strong>in</strong>g and concentrate of ‘simple’ boundary conditions<br />
def<strong>in</strong>ed below. Neumann and Dirichlet boundary conditions will be needed<br />
for this text only <strong>in</strong> chapter 6 <strong>in</strong> the proof of Lifshitz tails.<br />
For our purpose the most important feature of Neumann and Dirichlet boundary<br />
conditions is the so called Dirichlet-Neumann bracket<strong>in</strong>g . Suppose M 1 and M 2<br />
are disjo<strong>in</strong>t open sets <strong>in</strong> R d and M = (M 1 ∪ M 2 ) ◦ , ( ◦ denot<strong>in</strong>g the <strong>in</strong>terior) then<br />
−∆ N M 1<br />
⊕ −∆ N M 2<br />
≤ −∆ N M ≤ −∆ D M ≤ −∆ D M 1<br />
⊕ −∆ D M 2<br />
. (5.24)<br />
<strong>in</strong> the sense of quadratic forms. In particular the eigenvalues of the opera<strong>to</strong>rs <strong>in</strong><br />
(5.24) are <strong>in</strong>creas<strong>in</strong>g from left <strong>to</strong> right.<br />
We recall that a bounded opera<strong>to</strong>r A on a Hilbert space H is called positive (or<br />
positive def<strong>in</strong>ite or A ≥ 0) if<br />
〈ϕ, A ϕ〉 ≥ 0 for all ϕ ∈ H . (5.25)<br />
For unbounded A the validity of equation 5.25 is required for the (form-)doma<strong>in</strong><br />
of A only.<br />
By A ≤ B for two opera<strong>to</strong>rs A and B we mean B − A ≥ 0.<br />
For the lattice case we <strong>in</strong>troduce boundary conditions which we call Dirichlet and<br />
Neumann conditions as well. Our choice is guided by the cha<strong>in</strong> of <strong>in</strong>equalities(5.24).
36<br />
The easiest version of boundary conditions for the lattice is given by the follow<strong>in</strong>g<br />
procedure<br />
DEFINITION 5.15. The Laplacian with simple boundary conditions on Λ ⊂ Z d is<br />
the opera<strong>to</strong>r on l 2 (Λ) def<strong>in</strong>ed by<br />
(H 0 ) Λ (n, m) = 〈δ n , H 0 δ m 〉 (5.26)<br />
whenever both n and m belong <strong>to</strong> Λ. We also set H Λ = (H 0 ) Λ + V .<br />
In particular, if Λ is f<strong>in</strong>ite, the opera<strong>to</strong>r H Λ acts on a f<strong>in</strong>ite dimensional space, i.e.<br />
is a matrix.<br />
We are go<strong>in</strong>g <strong>to</strong> use simple boundary conditions frequently <strong>in</strong> this work. At a first<br />
glance simple boundary conditions seem <strong>to</strong> be a reasonable analog of Dirichlet<br />
boundary conditions. However, they do not satisfy (5.24) as we will see later.<br />
Thus, we will have <strong>to</strong> search for other boundary conditions.<br />
Let us def<strong>in</strong>e<br />
∂Λ = { (n, m) ∈ Z d × Z d ∣ ∣ || n − m|| 1 = 1 and<br />
either n ∈ Λ, m ∉ Λ or n ∉ Λ, m ∈ Λ } . (5.27)<br />
The set ∂Λ is the boundary of Λ. It consists of the edges connect<strong>in</strong>g po<strong>in</strong>ts <strong>in</strong> Λ<br />
with po<strong>in</strong>ts outside Λ. We also def<strong>in</strong>e the <strong>in</strong>ner boundary of Λ by<br />
∂ − Λ = { n ∈ Z d ∣ ∣ n ∈ Λ, ∃ m ∉ Λ (n, m) ∈ ∂Λ } (5.28)<br />
and the outer boundary by<br />
∂ + Λ = { m ∈ Z d ∣ ∣ m ∉ Λ, ∃ n ∈ Λ (n, m) ∈ ∂Λ } . (5.29)<br />
Hence ∂ + Λ = ∂ − (∁Λ) and the boundary ∂Λ consists of edges between ∂ − Λ and<br />
∂ + Λ.<br />
For any set Λ we def<strong>in</strong>e the boundary opera<strong>to</strong>r Γ Λ by<br />
Γ Λ (n, m) =<br />
{<br />
−1 if (n, m) ∈ ∂Λ,<br />
0 otherwise.<br />
Thus for the Hamili<strong>to</strong>nian H = H 0 + V we have the important relation<br />
(5.30)<br />
H = H Λ ⊕ H ∁Λ + Γ Λ . (5.31)<br />
In this equation we identified l 2 (Z d ) with l 2 (Λ) ⊕ l 2 (∁Λ).<br />
More precisely
37<br />
⎧<br />
⎨ H Λ (n, m) if n, m ∈ Λ,<br />
(H Λ ⊕ H ∁Λ )(n, m) = H<br />
⎩ ∁Λ (n, m) if n, m ∉ Λ,<br />
0 otherwise.<br />
(5.32)<br />
In other words H Λ ⊕ H ∁Λ is a block diagonal matrix and Γ Λ is the part of H which<br />
connects these blocks.<br />
It is easy <strong>to</strong> see, that Γ Λ is neither negative nor positive def<strong>in</strong>ite. Consequently, the<br />
opera<strong>to</strong>r H Λ will not satisfy any <strong>in</strong>equality of the type (5.24).<br />
To obta<strong>in</strong> analogs <strong>to</strong> Dirichlet and Neumann boundary conditions we should substitute<br />
the opera<strong>to</strong>r Γ Λ <strong>in</strong> (5.31) by a negative def<strong>in</strong>ite resp. positive def<strong>in</strong>ite opera<strong>to</strong>r<br />
and H Λ ⊕ H ∁Λ by an appropriate block diagonal matrix.<br />
For the opera<strong>to</strong>r H 0 the diagonal term H 0 (i, i) = 2d gives the number of sites j<br />
<strong>to</strong> which i is connected (namely the 2d neighbors <strong>in</strong> Z d ). This number is called<br />
the coord<strong>in</strong>ation number of the graph Z d . In the matrix H Λ the edges <strong>to</strong> ∁Λ are<br />
removed but the diagonal still conta<strong>in</strong>s the ‘old’ number of adjacent edges. Let us<br />
set n Λ (i) = | {j ∈ Λ| || j − i|| 1 = 1}| <strong>to</strong> be the number of sites adjacent <strong>to</strong> i <strong>in</strong> Λ,<br />
the coord<strong>in</strong>ation number for the graph Λ. n Λ (i) = 2d as long as i ∈ Λ\∂ − Λ but<br />
n Λ (i) < 2d at the boundary. We also def<strong>in</strong>e the adjacency matrix on Λ by<br />
{<br />
−1 if i, j ∈ Λ, || i − j||1 = 1<br />
A Λ (i, j) =<br />
0 otherwise.<br />
(5.33)<br />
The opera<strong>to</strong>r (H 0 ) Λ on l 2 (Λ) is given by<br />
where 2d denotes a multiple of the identity.<br />
(H 0 ) Λ = 2d + A Λ (5.34)<br />
DEFINITION 5.16. The Neumann Laplacian on Λ ⊂ Z d is the opera<strong>to</strong>r on l 2 (Λ)<br />
def<strong>in</strong>ed by<br />
(H 0 ) N Λ = n Λ + A Λ . (5.35)<br />
Above n Λ stands for the multiplication opera<strong>to</strong>r with the function n Λ (i) on l 2 (Λ).<br />
REMARK 5.17.<br />
(1) In (H 0 ) Λ the off diagonal term ‘connect<strong>in</strong>g’ Λ <strong>to</strong> Z d \Λ are removed.<br />
However, through the diagonal term 2d the opera<strong>to</strong>r still ‘remembers’<br />
there were 2d neighbors orig<strong>in</strong>ally.<br />
(2) The Neumann Laplacian H N 0 on Λ is also called the graph Laplacian. It<br />
is the canonical and <strong>in</strong>tr<strong>in</strong>sic Laplacian with respect <strong>to</strong> the graph structure<br />
of Λ. It ‘forgets’ completely that the set Λ is imbedded <strong>in</strong> Z d .<br />
(3) The quadratic form correspond<strong>in</strong>g <strong>to</strong> (H 0 ) N Λ<br />
is given by<br />
〈u, (H 0 ) N Λ v〉 = 1 ∑<br />
(u(n) − u(m))(v(n) − v(m)) .<br />
2<br />
n,m∈Λ<br />
|| n−m|| 1 =1
38<br />
DEFINITION 5.18. The Dirichlet Laplacian on Λ is the opera<strong>to</strong>r on l 2 (Λ) def<strong>in</strong>ed<br />
by<br />
REMARK 5.19.<br />
(H 0 ) D Λ = 2d + (2d − n Λ ) + A Λ .<br />
(1) The def<strong>in</strong>ition of the Dirichlet Laplacian may look a bit strange at the<br />
first glance. The ma<strong>in</strong> motivation for this def<strong>in</strong>ition is <strong>to</strong> preserve the<br />
properties (5.24) of the cont<strong>in</strong>uous analog.<br />
(2) The Dirichlet Laplacian not only remembers that there were 2d neighbor<strong>in</strong>g<br />
sites before <strong>in</strong>troduc<strong>in</strong>g boundary conditions, it even <strong>in</strong>creases the<br />
diagonal entry by one for each adjacent edge which was removed. Very<br />
loosely speak<strong>in</strong>g, one might say that the po<strong>in</strong>ts at the boundary get an<br />
additional connection for every ‘miss<strong>in</strong>g’ l<strong>in</strong>k <strong>to</strong> po<strong>in</strong>ts outside Λ.<br />
It is not hard <strong>to</strong> see, that<br />
and<br />
(H 0 ) N Λ ≤ (H 0 ) Λ ≤ (H 0 ) D Λ (5.36)<br />
H 0 = (H 0 ) N Λ ⊕ (H 0 ) N ∁Λ + ΓN Λ (5.37)<br />
= (H 0 ) D Λ ⊕ (H 0 ) D ∁Λ + ΓD Λ (5.38)<br />
with<br />
and<br />
The opera<strong>to</strong>r Γ N Λ<br />
⎧<br />
2d − n ⎪⎨ Λ (i) if i = j, i ∈ Λ,<br />
Γ N 2d − n<br />
Λ (i, j) =<br />
∁Λ (i) if i = j, i ∈ ∁Λ,<br />
−1 if (i, j) ∈ ∂Λ,<br />
⎪⎩<br />
0 otherwise<br />
⎧<br />
n ⎪⎨ Λ (i) − 2d if i = j, i ∈ Λ,<br />
Γ D n<br />
Λ (i, j) = ∁Λ (i) − 2d if i = j, i ∈ ∁Λ,<br />
−1 if (i, j) ∈ ∂Λ,<br />
⎪⎩<br />
0 otherwise.<br />
is positive def<strong>in</strong>ite as<br />
(5.39)<br />
(5.40)<br />
〈u, Γ N Λ v〉 = 1 ∑ ( )( )<br />
u(i) − u(j) v(i) − v(j)<br />
2<br />
(i,j)∈∂Λ<br />
is its quadratic form. In a similar way, we see that Γ D Λ<br />
is negative def<strong>in</strong>ite, s<strong>in</strong>ce<br />
〈u, Γ D Λ v〉 = − 1 ∑ ( )( )<br />
u(i) + u(j) v(i) + v(j)<br />
2<br />
(i,j)∈∂Λ<br />
Hence we have <strong>in</strong> analogy <strong>to</strong> (5.24)<br />
(H 0 ) N Λ ⊕ (H 0 ) N ∁Λ ≤ H 0 ≤ (H 0 ) D Λ ⊕ (H 0 ) D ∁Λ . (5.41)
If H = H 0 + V we def<strong>in</strong>e H Λ = (H 0 ) Λ + V , where <strong>in</strong> the latter expression<br />
V stands for the multiplication with the function V restricted <strong>to</strong> Λ. Similarly,<br />
HΛ N = (H 0) N Λ + V and HD Λ = (H 0) D Λ + V .<br />
These opera<strong>to</strong>rs satisfy<br />
H N Λ ⊕ H N ∁Λ ≤ H ≤ HD Λ ⊕ H D ∁Λ . (5.42)<br />
39<br />
For Λ 1 ⊂ Λ ⊂ Z d we have analogs of the ‘splitt<strong>in</strong>g’ formulae (5.31), (5.37) and<br />
(5.38). To formulate them it will be useful <strong>to</strong> def<strong>in</strong>e the ‘relative’ boundary ∂ Λ2 Λ 1<br />
of Λ 1 ⊂ Λ 2 <strong>in</strong> Λ 2 .<br />
∂ Λ2 Λ 1 = ∂Λ 1 ∩ (Λ 2 × Λ 2 ) = ∂Λ 1 \ ∂Λ 2 (5.43)<br />
= { (i, j) ∣ ∣ || i − j|| 1 = 1 and i ∈ Λ 1 , j ∈ Λ 2 \ Λ 1 or i ∈ Λ 2 \ Λ 1 , j ∈ Λ 1<br />
}<br />
The analogs of the splitt<strong>in</strong>g formulae are<br />
with<br />
H Λ2 = H Λ1 ⊕ H Λ2 \Λ 1<br />
+ Γ Λ 2<br />
Λ 1<br />
(5.44)<br />
H N Λ 2<br />
= H N Λ 1<br />
⊕ H N Λ 2 \Λ 1<br />
+ Γ Λ 2 N<br />
Λ 1<br />
(5.45)<br />
H D Λ 2<br />
= H D Λ 1<br />
⊕ H D Λ 2 \Λ 1<br />
+ Γ Λ 2 D<br />
Λ 1<br />
(5.46)<br />
⎧<br />
⎪⎨<br />
Γ Λ 2<br />
Λ 1<br />
(i, j) =<br />
⎪⎩<br />
⎧<br />
⎪⎨<br />
Λ 1<br />
(i, j) =<br />
⎪⎩<br />
Γ Λ 2 N<br />
⎧<br />
⎪⎨<br />
Λ 1<br />
(i, j) =<br />
⎪⎩<br />
Γ Λ 2 D<br />
0 if i = j and i ∈ Λ 1<br />
0 if i = j and i ∈ Λ 2 \ Λ 1<br />
−1 if (i, j) ∈ ∂ Λ2 Λ 1<br />
0 otherwise.<br />
n Λ2 (i) − n Λ1 (i) if i = j and i ∈ Λ 1<br />
n Λ2 (i) − n Λ2 \Λ 1<br />
(i) if i = j and i ∈ Λ 2 \ Λ 1<br />
−1 if (i, j) ∈ ∂ Λ2 Λ 1<br />
0 otherwise.<br />
n Λ1 (i) − n Λ2 (i) if i = j and i ∈ Λ 1<br />
n Λ2 \Λ 1<br />
(i) − n Λ2 (i) if i = j and i ∈ Λ 2 \ Λ 1<br />
−1 if (i, j) ∈ ∂ Λ2 Λ 1<br />
0 otherwise.<br />
(5.47)<br />
(5.48)<br />
(5.49)<br />
In particular, for Λ = Λ 1 ∪ Λ 2 with disjo<strong>in</strong>t sets Λ 1 and Λ 2 we have<br />
H N Λ 1<br />
⊕ H N Λ 2<br />
≤ H N Λ ≤ H D Λ ≤ H D Λ 1<br />
⊕ H D Λ 2<br />
. (5.50)<br />
s<strong>in</strong>ce Γ Λ N<br />
Λ 1<br />
≥ 0 and Γ Λ D<br />
Λ 1<br />
≤ 0.
40<br />
5.3. The geometric resolvent equation.<br />
The equations (5.44), (5.45)and (5.46) allow us <strong>to</strong> prove the so called geometric<br />
resolvent equation. It expresses the resolvent of an opera<strong>to</strong>r on a larger set <strong>in</strong> terms<br />
of opera<strong>to</strong>rs on smaller sets. This equality is a central <strong>to</strong>ol of multiscale analysis.<br />
We do the calculations for simple boundary conditions (5.26) but the results are<br />
valid for Neumann and Dirichlet boundary conditions with the obvious changes.<br />
We start from equation (5.44) for Λ 1 ⊂ Λ 2 ⊂ Z d .<br />
For z ∈ C \ R this equation and the resolvent equation (3.18) imply<br />
(H Λ2 − z) −1<br />
= (H Λ1 ⊕ H Λ2 \Λ 1<br />
− z) −1 − (H Λ1 ⊕ H Λ2 \Λ 1<br />
− z) −1 Γ Λ 2<br />
Λ 1<br />
(H Λ2 − z) −1<br />
= (H Λ1 ⊕ H Λ2 \Λ 1<br />
− z) −1 − (H Λ2 − z) −1 Γ Λ 2<br />
Λ 1<br />
(H Λ1 ⊕ H Λ2 \Λ 1<br />
− z) −1 .<br />
In fact, (5.51) holds for z ∉ σ(H Λ1 ) ∪ σ(H Λ2 ) ∪ σ(H Λ2 \Λ 1<br />
).<br />
For n ∈ Λ 1 , m ∈ Λ 2 \Λ 1 we have<br />
(5.51)<br />
hence<br />
H Λ1 ⊕ H Λ2 \Λ 1<br />
(n, m) = 0<br />
(H Λ1 ⊕ H Λ2 \Λ 1<br />
− z) −1 (n, m) = 0 .<br />
Note that (H Λ1 ⊕ H Λ2 \Λ 1<br />
− z) −1 = (H Λ2 − z) −1 ⊕ (H Λ2 \Λ 1<br />
− z) −1 .<br />
Thus (5.51) gives (for n ∈ Λ 1 , m ∈ Λ 2 \Λ 1 )<br />
(H Λ2 − z) −1 (n, m)<br />
= − ∑<br />
(H Λ1 ⊕ H Λ2 − z) −1 (n, k) Γ Λ 2<br />
Λ 1<br />
(k, k ′ ) (H Λ2 − z) −1 (k ′ , m)<br />
k,k ′ ∈Λ 2<br />
∑<br />
=<br />
(H Λ1 − z) −1 (n, k) (H Λ2 − z) −1 (k ′ , m) . (5.52)<br />
(k,k ′ )∈∂Λ 1<br />
k∈Λ 1 , k ′ ∈Λ 2<br />
We summarize <strong>in</strong> the follow<strong>in</strong>g theorem<br />
THEOREM 5.20 (Geometric resolvent equation).<br />
If Λ 1 ⊂ Λ 2 and n ∈ Λ 1 , m ∈ Λ 2 \ Λ 1 and if z ∉ ( σ(H Λ1 ) ∪ σ(H Λ2 ) ) , then<br />
=<br />
(H Λ2 − z) −1 (n, m)<br />
∑<br />
(H Λ1 − z) −1 (n, k) (H Λ2 − z) −1 (k ′ , m) . (5.53)<br />
(k,k ′ )∈∂Λ 1<br />
k∈Λ 1 , k ′ ∈Λ 2<br />
Equation (5.53) is the geometric resolvent equation. It expresses the resolvent on<br />
a large set (Λ 2 ) <strong>in</strong> terms of the resolvent on a smaller set (Λ 1 ). Of course, the right<br />
hand side still conta<strong>in</strong>s the resolvent on the large set.
REMARK 5.21. Above we derived (5.53) only for z ∉ σ(H Λ2 \Λ 1<br />
). However, both<br />
sides of (5.53) exist and are analytic outside σ(H Λ1 ) ∪ σ(H Λ2 ), so the formula is<br />
valid for all z outside σ(H Λ1 ) ∪ σ(H Λ2 )<br />
We <strong>in</strong>troduce a short-hand notation for the matrix elements of resolvents<br />
G Λ z (n, m) = (H Λ − z) −1 (n, m). (5.54)<br />
The functions G Λ z are called Green’s functions .<br />
With this notation the geometric resolvent equation reads<br />
∑<br />
G Λ 2<br />
z (n, m) = G Λ 1<br />
z (n, k)G Λ 2<br />
z (k ′ , m) . (5.55)<br />
(k,k ′ )∈∂Λ 1<br />
k∈Λ 1 , k ′ ∈Λ 2<br />
There are analogous equations <strong>to</strong> (5.53) for Dirichlet or Neumann boundary conditions<br />
which can be derived from (5.46) and (5.46) <strong>in</strong> the same way as above.<br />
5.4. <strong>An</strong> alternative approach <strong>to</strong> the density of states.<br />
In this section we present an alternative def<strong>in</strong>ition of the density of states measure.<br />
Perhaps, this is the more traditional one. We prove its equivalence <strong>to</strong> the def<strong>in</strong>ition<br />
given above.<br />
In section 5.1 we def<strong>in</strong>ed the density of states measure by start<strong>in</strong>g with a function<br />
ϕ of the Hamil<strong>to</strong>nian, tak<strong>in</strong>g its trace restricted <strong>to</strong> a cube Λ L and normaliz<strong>in</strong>g<br />
this trace. In the second approach we first restrict the Hamil<strong>to</strong>nian H ω <strong>to</strong> Λ L with<br />
appropriate boundary conditions, apply the function ϕ <strong>to</strong> the restricted Hamil<strong>to</strong>nian<br />
and then take the normalized trace.<br />
For any Λ let H X Λ be either H Λ or H N Λ or HD Λ . We def<strong>in</strong>e the measures ˜νX L ( i.e.<br />
˜ν L , ˜ν N L , ˜νD L ) by<br />
∫<br />
ϕ(λ) d˜ν X L (λ) = 1<br />
|Λ L | tr ϕ(HX Λ L<br />
) . (5.56)<br />
Note that the opera<strong>to</strong>rs H X Λ L<br />
act on the f<strong>in</strong>ite dimensional Hilbert space l 2 (Λ L ), so<br />
their spectra consist of eigenvalues E n (H X Λ L<br />
) which we enumerate <strong>in</strong> <strong>in</strong>creas<strong>in</strong>g<br />
order<br />
E 0 (H X Λ L<br />
) ≤ E 1 (H X Λ L<br />
) ≤ . . . .<br />
In this enumeration we repeat each eigenvalue accord<strong>in</strong>g <strong>to</strong> its multiplicity (see<br />
also (3.43).<br />
With this notation (5.56) reads<br />
∫<br />
ϕ(λ) d˜ν L X (λ) = 1 ∑<br />
ϕ(E n (HΛ X |Λ L |<br />
L<br />
)) .<br />
The measure ˜ν<br />
L X is concentrated on the eigenvalues of HX Λ L<br />
. If E is an eigenvalue<br />
of HΛ X L<br />
then ˜ν<br />
L X ({E}) is equal <strong>to</strong> the dimension of the eigenspace correspond<strong>in</strong>g<br />
<strong>to</strong> E.<br />
n<br />
41
42<br />
We def<strong>in</strong>e the eigenvalue count<strong>in</strong>g function by<br />
N(HΛ X , E) = ∣ ∣{n | E n (HΛ X ) < E} ∣ (5.57)<br />
(where |M| is the number of elements of M). Then 1<br />
|Λ L | N(HX Λ L<br />
, E) is the distribution<br />
function of the measure ˜ν<br />
L X , i. e.<br />
1<br />
|Λ L | N(HX Λ L<br />
, E) =<br />
∫<br />
χ (−∞,E) (λ) d ˜ν X L (λ) . (5.58)<br />
THEOREM 5.22. The measures ˜ν L , ˜ν<br />
L D and ˜νN L<br />
<strong>to</strong> the density of states measure ν.<br />
converge P-almost surely vaguely<br />
PROOF: We give the proof for ˜ν L . <strong>An</strong> easy modification gives the result for ˜ν<br />
L<br />
D<br />
and ˜ν<br />
L N as well. To prove that ˜ν L converges vaguely <strong>to</strong> ν it suffices <strong>to</strong> prove<br />
∫<br />
∫<br />
ϕ(λ) d˜ν L (λ) → ϕ(λ) dν(λ)<br />
for all ϕ of the form<br />
ϕ(x) = r z (x) = 1<br />
x − z<br />
for z ∈ C \ R<br />
because l<strong>in</strong>ear comb<strong>in</strong>ation of these functions are dense <strong>in</strong> C ∞ (R) by the S<strong>to</strong>ne-<br />
Weierstraß Theorem (see Section 3.3). ( C ∞ (R) are the cont<strong>in</strong>uous functions vanish<strong>in</strong>g<br />
at <strong>in</strong>f<strong>in</strong>ity.)<br />
We have<br />
∫<br />
r z (λ) d˜ν L (λ) =<br />
1<br />
(2L + 1) d tr ( (H ΛL −z) −1) =<br />
1 ∑<br />
(2L + 1) d<br />
n∈Λ L<br />
(H ΛL −z) −1 (n, n)<br />
and<br />
∫<br />
r z (λ) dν L (λ) =<br />
1<br />
(2L + 1) d tr ( χ ΛL (H−z) −1) =<br />
1 ∑<br />
(2L + 1) d<br />
n∈Λ L<br />
(H−z) −1 (n, n)<br />
We use the resolvent equation <strong>in</strong> the form (5.51) for n ∈ Λ L :
43<br />
∣ ∑<br />
(H ΛL − z) −1 (n, n) − (H − z) −1 (n, n) ∣ n∈Λ L<br />
= ∣ ∑ ∑<br />
(H ΛL − z) −1 (n, k) (H − z) −1 (k ′ , n) ∣ n∈Λ L (k,k ′ )∈∂Λ L<br />
k∈Λ L , k ′ ∈∁Λ L<br />
∑<br />
≤<br />
|(H ΛL − z) −1 (n, k)| 2) 12 · ( ∑<br />
|(H − z) −1 (k ′ , n)| 2) 1 2<br />
=<br />
( ∑<br />
(k,k ′ )∈∂Λ L n<br />
k∈Λ L , k ′ ∈∁Λ L<br />
∑<br />
||(H ΛL − z) −1 δ k || · ||(H − z) −1 δ k ′||<br />
(k,k ′ )∈∂Λ L<br />
k∈Λ L , k ′ ∈∁Λ L<br />
≤ c L d−1 ||(H ΛL − z) −1 || · ||(H − z) −1 ||<br />
≤<br />
c<br />
(Im z) 2 Ld−1 . (5.59)<br />
Hence<br />
n<br />
∫<br />
|<br />
∫<br />
r z (λ) d˜ν L (λ) −<br />
r z (λ) dν L (λ)| ≤<br />
c ′<br />
(Im z) 2 · 1<br />
L → 0 as L → ∞ .<br />
□<br />
5.5. The Wegner estimate.<br />
We cont<strong>in</strong>ue with the celebrated ‘Wegner estimate’. This result due <strong>to</strong> Wegner<br />
[140] shows not only the regularity of the density of states, it is also a key <strong>in</strong>gredient<br />
<strong>to</strong> prove <strong>An</strong>derson localization. We set N Λ (E) := N(H Λ , E).<br />
THEOREM 5.23. (Wegner estimate ) Suppose the measure P 0 has a bounded density<br />
g, (i.e. P 0 (A) = ∫ A g(λ)dλ, ||g|| ∞ < ∞) then<br />
E ( N Λ (E + ε) − N Λ (E − ε) ) ≤ C ‖ g ‖ ∞ |Λ| ε . (5.60)<br />
Before we prove this estimate we note two important consequences.<br />
COROLLARY 5.24. Under the assumption of Theorem 5.23 the <strong>in</strong>tegrated density<br />
of states is absolutely cont<strong>in</strong>uous with a bounded density n(E).<br />
Thus N(E) = ∫ E<br />
−∞<br />
n(λ) dλ. We call n(λ) the density of states. Sometimes, we<br />
also call N the density of states, which, we admit, is an abuse of language.<br />
COROLLARY 5.25. Under the assumptions of Theorem 5.23 we have for any E<br />
and Λ<br />
P ( dist(E, σ(H Λ )) < ε ) ≤ C ‖ g ‖ ∞ ε |Λ| . (5.61)
44<br />
PROOF (Corollary 5.25) : By the Chebycheff <strong>in</strong>equality we get<br />
P ( dist(E, σ(H Λ )) < ε )<br />
= P ( N Λ (E + ε) − N Λ (E − ε) ≥ 1 )<br />
≤ E ( N Λ (E + ε) − N Λ (E − ε) )<br />
≤ C ‖ g ‖ ∞ ε |Λ| by Theorem (5.23). (5.62)<br />
□<br />
PROOF (Corollary 5.24) :<br />
N(E + ε) − N(E − ε) =<br />
We turn <strong>to</strong> the proof of the theorem.<br />
By Theorem 5.23 we have<br />
lim<br />
|Λ|→∞<br />
1<br />
|Λ| E ( N Λ (E + ε) − N Λ (E − ε) )<br />
≤ C ‖ g ‖ ∞ ε . (5.63)<br />
PROOF (Wegner estimate) : Let ϱ be a non decreas<strong>in</strong>g C ∞ -function with<br />
ϱ(λ) = 1 for λ ≥ ε, ϱ(λ) = 0 for λ ≤ −ε and consequently 0 ≤ ϱ(λ) ≤ 1. Then<br />
hence<br />
Consequently,<br />
0 ≤ χ (−∞,E+ε) (λ) − χ (−∞,E−ε) (λ)<br />
≤ ϱ(λ − E + 2ε) − ϱ(λ − E − 2ε)<br />
0 ≤ χ (−∞,E+ε) (H Λ ) − χ (−∞,E−ε) (H Λ )<br />
≤<br />
ϱ(H Λ − E + 2ε) − ϱ(H Λ − E − 2ε).<br />
N Λ (E + ε) − N Λ (E − ε)<br />
= tr ( χ (−∞,E+ε) (H Λ ) − χ (−∞,E−ε) (H Λ ) )<br />
≤ tr ϱ(H Λ − E + 2ε) − tr ϱ(H Λ − E − 2ε) . (5.64)<br />
To compute the expectation of (5.64) we look upon the opera<strong>to</strong>rs H Λ (and their<br />
eigenvalues E n (H Λ )) as functions of the values V Λ = {V i } i∈Λ of the potential<br />
<strong>in</strong>side Λ. More precisely, we view the mapp<strong>in</strong>g<br />
V Λ → H Λ = H Λ (V Λ )<br />
as a matrix-valued function on R |Λ| . This function is differentiable and<br />
□<br />
The function<br />
( ∂HΛ<br />
∂V i<br />
)lm<br />
(E, V Λ ) → tr ϱ ( H Λ (V Λ ) − E ) .<br />
= δ lm δ li . (5.65)
45<br />
is differentiable as well. Furthermore, s<strong>in</strong>ce<br />
H Λ (V Λ ) − E = H Λ (V Λ − E) . (5.66)<br />
it follows<br />
tr ϱ ( H Λ (V Λ ) − E ) = F ( {V i − E} i∈Λ<br />
)<br />
(5.67)<br />
and consequently<br />
∂<br />
∂E<br />
(<br />
tr ϱ ( H Λ (V Λ ) − E )) = − ∑ i∈Λ<br />
∂<br />
(<br />
tr ϱ ( H Λ (V Λ ) − E )) (5.68)<br />
∂V i<br />
Therefore, with (5.64)<br />
N Λ (E + ε) − N Λ (E − ε) ≤ tr ϱ (H Λ − E + 2ε) − tr ϱ (H Λ − E − 2ε)<br />
= − ( tr ϱ (H Λ − (E + 2ε)) − tr ϱ( H Λ − (E − 2ε)) )<br />
= −<br />
=<br />
∫ E+2ε<br />
E−2ε<br />
∫ E+2ε<br />
E−2ε<br />
∂<br />
∂η<br />
∑<br />
j∈Λ<br />
(<br />
ϱ ( H Λ (V Λ ) − η )) dη<br />
∂<br />
∂V j<br />
tr ϱ ( H Λ (V Λ − η) ) dη. (5.69)<br />
Therefore<br />
E ( N Λ (E + ε) − N Λ (E − ε) )<br />
≤ E ( ∑<br />
= ∑ j∈Λ<br />
∫ E+2ε<br />
j∈Λ<br />
E−2ε<br />
∫ E+2ε<br />
E−2ε<br />
∂<br />
∂V j<br />
tr (ϱ (H Λ (V Λ ) − η) ) dη)<br />
( ∂<br />
E tr ( ϱ (H Λ (V Λ ) − η) )) dη. (5.70)<br />
∂V j<br />
S<strong>in</strong>ce the random variables V ω (i) are <strong>in</strong>dependent and have the common distribution<br />
dP 0 (V i ) = g(V i ) dV i , the expectation E is just <strong>in</strong>tegration with respect <strong>to</strong><br />
the product of these distributions. Moreover, s<strong>in</strong>ce supp P 0 is compact the <strong>in</strong>tegral<br />
over the variable V i can be restricted <strong>to</strong> [−M, +M] for some M large enough.
46<br />
Hence<br />
( ∂<br />
E tr ( ϱ (H Λ (V Λ ) − η) ))<br />
∂V j<br />
=<br />
=<br />
∫ +M<br />
−M<br />
∫ +M<br />
−M<br />
. . .<br />
. . .<br />
∫ +M<br />
−M<br />
∫ +M<br />
−M<br />
∂<br />
tr ( ϱ(H Λ (V Λ ) − η) ) ∏<br />
∂V j<br />
g(V i ) ∏ dV i<br />
i∈Λ i∈Λ<br />
( ∫ +M<br />
∂<br />
tr ( ϱ(H Λ (V Λ ) − η) ) ) ∏<br />
g(V j ) dV j g(V i ) dV i .<br />
∂V j<br />
−M<br />
S<strong>in</strong>ce tr ( ϱ(H Λ (V Λ ) − η) ) is non decreas<strong>in</strong>g <strong>in</strong> V j we can estimate<br />
i∈Λ<br />
i≠j<br />
(5.71)<br />
∫ +M<br />
−M<br />
∂<br />
∂V j<br />
tr ( ϱ(H Λ (V Λ ) − η) ) g(V j ) dV j<br />
≤ ||g|| ∞<br />
(<br />
tr ( ϱ(H Λ (V Λ , V j = M) − η) ) − tr ( ϱ(H Λ (V Λ , V j = −M) − η) ))<br />
(5.72)<br />
where H Λ (V Λ , V j<br />
potential<br />
= a) = H 0Λ + Ṽ is the <strong>An</strong>derson Hamil<strong>to</strong>nian on Λ with<br />
Ṽ i =<br />
{<br />
Vi for i ≠ j<br />
a for i = j.<br />
(5.73)<br />
To estimate the right hand side of <strong>in</strong>equality (5.72) we will use the follow<strong>in</strong>g<br />
Lemma:<br />
LEMMA 5.26. Let A be a selfadjo<strong>in</strong>t opera<strong>to</strong>r bounded below with purely discrete<br />
spectrum and eigenvalues E 0 ≤ E 1 ≤ . . . repeated accord<strong>in</strong>g <strong>to</strong> multiplicity. If<br />
B is a symmetric positive rank one opera<strong>to</strong>r then à = A + B has eigenvalue E˜<br />
n<br />
with E n ≤ E ˜ n ≤ E n+1 .<br />
Given the Lemma we cont<strong>in</strong>ue the proof of the theorem.<br />
We set A = H Λ (V Λ , V j = −M) and à = H Λ(V Λ , V j = +M). Obviously their<br />
difference is a (positive) rank one opera<strong>to</strong>r<br />
=<br />
≤<br />
≤<br />
tr ϱ(Ã − η) − tr ϱ(A − η)<br />
∑ (<br />
ϱ( En ˜ − η) − ϱ(E n − η) )<br />
n<br />
∑<br />
n<br />
sup<br />
λ,µ<br />
(<br />
ϱ(En+1 − η) − ϱ(E n − η) )<br />
ϱ(λ) − ϱ(µ)<br />
= 1 . (5.74)
47<br />
Thus from (5.72) we have<br />
S<strong>in</strong>ce ∫ +M<br />
−M<br />
∫ +M<br />
−M<br />
So, (5.70) implies<br />
∂<br />
∂V j<br />
tr ( ϱ(H Λ (V Λ ) − η) ) g(V j ) dV j ≤ || g|| ∞ .<br />
g(v)dv = 1 we conclude from (5.71) and (5.72) that<br />
( ∂<br />
E tr ( ϱ (H Λ (V Λ ) − η) )) ≤ || g|| ∞ .<br />
∂V j<br />
E ( N Λ (E + ε) − N Λ (E − ε) ) ≤ 4 || g|| ∞ | Λ| ε . (5.75)<br />
□<br />
PROOF (Lemma) : S<strong>in</strong>ce B is a positive symmetric rank one opera<strong>to</strong>r it is of<br />
the form B = c |h〉〈h| with c ≥ 0, i.e. B ϕ = c 〈h, ϕ〉 h for some h.<br />
By the m<strong>in</strong>-max pr<strong>in</strong>ciple (Theorem 3.1)<br />
Ẽ n = sup<br />
ψ 1 ,...,ψ n−1<br />
≤<br />
≤<br />
≤<br />
sup<br />
ψ 1 ,...,ψ n−1<br />
sup<br />
ψ 1 ,...,ψ n−1<br />
sup<br />
ψ 1 ,...,ψ n−1 ,ψ n<br />
<strong>in</strong>f<br />
ϕ⊥ψ 1 ,...,ψ n−1<br />
||ϕ||=1<br />
<strong>in</strong>f<br />
ϕ⊥ψ 1 ,...,ψ n−1<br />
ϕ⊥h ||ϕ||=1<br />
<strong>in</strong>f<br />
ϕ⊥ψ 1 ,...,ψ n−1 ,h<br />
||ϕ||=1<br />
<strong>in</strong>f<br />
ϕ⊥ψ 1 ,...,ψ n−1 ,ψn<br />
||ϕ||=1<br />
〈ϕ, A ϕ〉 + c |〈ϕ, h〉| 2<br />
〈ϕ, A ϕ〉 + c |〈ϕ, h〉| 2<br />
〈ϕ, A ϕ〉<br />
〈ϕ, A ϕ〉<br />
= E n+1 . (5.76)<br />
By the Wegner estimate we know that any given energy E is not an eigenvalue of<br />
(H ω ) ΛL for almost all ω. On the other hand it is clear that for any given ω there<br />
are (as a rule |Λ L |) eigenvalues of (H ω ) ΛL .<br />
This simple fact illustrates that we are not allowed <strong>to</strong> <strong>in</strong>terchange ‘any given E’<br />
and ‘for P−almost all ω’ <strong>in</strong> assertions like the one above. What goes wrong is that<br />
we are try<strong>in</strong>g <strong>to</strong> take an uncountable union of sets of measure zero. This union may<br />
have any measure, if it is measurable at all.<br />
In the follow<strong>in</strong>g we demonstrate a way <strong>to</strong> overcome these difficulties (<strong>in</strong> a sense).<br />
This idea is extremely useful when we want <strong>to</strong> prove pure po<strong>in</strong>t spectrum.<br />
THEOREM 5.27. If Λ 1 , Λ 2 are disjo<strong>in</strong>t f<strong>in</strong>ite subsets of Z d , then<br />
P ( There is an E ∈ R such that dist(E, σ(H Λ1 )) < ε and dist(σ(E, H Λ2 ))) < ε )<br />
≤ 2 C ‖ g ‖ ∞ ε | Λ 1 || Λ 2 | .<br />
□
48<br />
We start the proof with the follow<strong>in</strong>g lemma.<br />
LEMMA 5.28. If Λ 1 , Λ 2 are disjo<strong>in</strong>t f<strong>in</strong>ite subsets of Z d then<br />
P ( dist(σ(H Λ1 ), σ(H Λ2 )) < ε ) ≤ C ‖ g ‖ ∞ ε |Λ 1 ||Λ 2 | .<br />
PROOF (Lemma) : S<strong>in</strong>ce Λ 1 ∩ Λ 2 = ∅ the random potentials <strong>in</strong> Λ 1 and Λ 2 are<br />
<strong>in</strong>dependent of each other and so are the eigenvalues E (1)<br />
0 ≤ E (1)<br />
1 ≤ ... of H Λ1<br />
and the eigenvalues E (2)<br />
0 ≤ E (2)<br />
1 ≤ ... of H Λ2 .<br />
We denote the probability (resp. the expectation) with respect <strong>to</strong> the random<br />
variables <strong>in</strong> Λ by P Λ (resp. E Λ ). S<strong>in</strong>ce the random variables {V ω (n)} n∈Λ1 and<br />
{V ω (n)} n∈Λ2 are <strong>in</strong>dependent for Λ 1 ∩ Λ 2 = ∅ we have that for such sets P Λ1 ∪Λ 2<br />
is the product measure P Λ1 ⊗ P Λ2 .<br />
We compute<br />
P ( dist(σ(Λ 1 ), σ(Λ 2 )) < ε ) = P ( m<strong>in</strong> dist(E (1)<br />
i<br />
i<br />
, σ(H Λ2 )) < ε )<br />
≤<br />
≤<br />
From Theorem 5.23 we know that<br />
|Λ 1 |<br />
∑<br />
i=1<br />
|Λ 1 |<br />
∑<br />
i=1<br />
|Λ 1 |<br />
∑<br />
≤<br />
i=1<br />
P ( dist(E (1)<br />
i<br />
, σ(H Λ2 )) < ε )<br />
P Λ1 ⊗ P Λ2<br />
(<br />
dist(E<br />
(1)<br />
i<br />
, σ(H Λ2 )) < ε )<br />
E Λ1<br />
(<br />
P Λ2<br />
(<br />
dist(E<br />
(1)<br />
i<br />
, σ(H Λ2 )) < ε ) ) .<br />
P Λ2<br />
(<br />
dist(E, σ(HΛ2 )) < ε ) ≤ C ‖ g ‖ ∞ ε |Λ 2 | .<br />
(5.77)<br />
Hence, we obta<strong>in</strong><br />
(5.77) ≤ C ‖ g ‖ ∞ ε |Λ 2 ||Λ 1 | .<br />
□<br />
The proof of the theorem is now easy.<br />
PROOF (Theorem) :<br />
P ( There is an E ∈ R such that dist(σ(H Λ1 ), E) < ε and dist(σ(H Λ2 , E))) < ε )<br />
≤ P ( dist(σ(H Λ1 ), σ(H Λ2 )) < 2 ε)<br />
≤ 2 C ‖ g ‖ ∞ ε |Λ 1 ||Λ 2 |<br />
by the lemma.<br />
□
49<br />
Notes and Remarks<br />
General references for the density of states are [109], [63], [10] and [36], [58]<br />
[121] and [138]. A thorough discussion of the geometric resolvent equation <strong>in</strong> the<br />
context of perturbation theory can be found <strong>in</strong> [40], [47] and [127].<br />
In the context of the discrete Laplacian Dirichlet and Neumann boundary conditions<br />
were <strong>in</strong>troduced and <strong>in</strong>vestigated <strong>in</strong> [121]. See also [68].<br />
For discrete ergodic opera<strong>to</strong>rs the <strong>in</strong>tegrated density of states N is log-Hölder cont<strong>in</strong>uous,<br />
see [29]. Our proof of the cont<strong>in</strong>uity of N is tailored after [36].<br />
For results concern<strong>in</strong>g the Wegner estimates see [140], [59], [130] [27], [26],and<br />
[138], as well as references given there.
51<br />
6.1. Statement of the Result.<br />
6. Lifshitz tails<br />
Already <strong>in</strong> the 1960s, the physicist I. Lifshitz observed that the low energy behavior<br />
of the density of states changes drastically if one <strong>in</strong>troduces disorder <strong>in</strong> a system.<br />
More precisely, Lifshitz found that<br />
N(E) ∼ C (E − E 0 ) d 2 E ↘ E 0 (6.1)<br />
for the ordered case (i.e. periodic potential), E 0 be<strong>in</strong>g the <strong>in</strong>fimum of the spectrum,<br />
and<br />
N(E) ∼ C 1 e −C (E−E 0) − d 2<br />
E ↘ E 0 (6.2)<br />
for the disordered case. The behavior (6.2) of N is now called Lifshitz behavior<br />
or Lifshitz tails. We will prove (a weak form of) Lifshitz tails for the <strong>An</strong>derson<br />
model. This result is an <strong>in</strong>terest<strong>in</strong>g and important result on its own. It is also used<br />
as an <strong>in</strong>put for the proof of <strong>An</strong>derson localization.<br />
If P 0 is the common distribution of the <strong>in</strong>dependent random variables V ω (i), we<br />
denote by a 0 the <strong>in</strong>fimum of the support supp P 0 of P 0 . From Theorem 3.9 we have<br />
E 0 = <strong>in</strong>f σ(H ω ) = a 0 P-almost surely. We assume that P 0 is not trivial, i.e. is<br />
not concentrated <strong>in</strong> a s<strong>in</strong>gle po<strong>in</strong>t. Moreover, we suppose that<br />
P 0 ([a 0 , a 0 + ɛ]) ≥ Cɛ κ , for some C, κ > 0 . (6.3)<br />
Under these assumptions we prove:<br />
THEOREM 6.1 (Lifshitz-tails).<br />
ln | ln N(E)|<br />
lim<br />
E↘E 0 ln(E − E 0 ) = −d 2 . (6.4)<br />
REMARK 6.2. (6.4) is a weak form of (6.2). The asymp<strong>to</strong>tic formula (6.2) suggests<br />
that we should expect at least<br />
lim<br />
E↘E 0<br />
ln N(E)<br />
(E − E 0 ) − d 2<br />
= − C . (6.5)<br />
Lifshitz tails can be proven <strong>in</strong> the strong form (6.5) for the Poisson random potential<br />
(see [38] and [108]). In general, however, there can be a logarithmic correction<br />
<strong>to</strong> (6.5) (see [102]) so that we can only expect the weak form (6.4). This form of<br />
the asymp<strong>to</strong>tics is called the ‘doublelogarithmic’ asymp<strong>to</strong>tics.<br />
To prove the theorem, we show an upper and a lower bound.
52<br />
6.2. Upper bound.<br />
For the upper bound we will need Temple’s <strong>in</strong>equality , which we state and prove<br />
for the reader’s convenience.<br />
LEMMA 6.3 (Temple’s <strong>in</strong>equality). Let A be a self-adjo<strong>in</strong>t opera<strong>to</strong>r and E 0 =<br />
<strong>in</strong>f σ(A) be an isolated non degenerate eigenvalue. We set E 1 = <strong>in</strong>f ( σ(A)\{E 0 } ) .<br />
If ψ ∈ D(A) with ||ψ|| = 1 satisfies<br />
〈ψ, Aψ〉 < E 1 ,<br />
then<br />
E 0 ≥ 〈ψ, Aψ〉 − 〈ψ, A2 ψ〉 − 〈ψ, Aψ〉 2<br />
E 1 − 〈ψ, Aψ〉<br />
.<br />
PROOF:<br />
By assumption we have<br />
(A − E 1 )(A − E 0 ) ≥ 0.<br />
Hence, for any ψ with norm 1<br />
〈ψ, A 2 ψ〉 − E 1 〈ψ, Aψ〉 − E 0 〈ψ, Aψ〉 + E 1 E 0 ≥ 0 .<br />
This implies<br />
E 1 E 0 − E 0 〈ψ, Aψ〉 ≥ E 1 〈ψ, Aψ〉 − 〈ψ, Aψ〉 2 − (〈ψ, A 2 ψ〉 − 〈ψ, Aψ〉 2 ) .<br />
S<strong>in</strong>ce E 1 − 〈ψ, Aψ〉 > 0, we obta<strong>in</strong><br />
E 0 ≥ 〈ψ, Aψ〉 − 〈ψ, A2 ψ〉 − 〈ψ, Aψ〉 2<br />
E 1 − 〈ψ, Aψ〉<br />
.<br />
□<br />
We proceed with the upper bound.<br />
PROOF (upper bound) :<br />
By add<strong>in</strong>g a constant <strong>to</strong> the potential we may assume that a 0 = <strong>in</strong>f supp (P 0 ) = 0,<br />
so that V ω (n) ≥ 0. By (5.50) we have that<br />
1<br />
N(E) ≤<br />
|Λ L | E ( N(HΛ N L<br />
, E) )<br />
≤ P ( E 0 (H N Λ L<br />
) < E ) (6.6)<br />
for any L, s<strong>in</strong>ce N(H N Λ L<br />
, E) ≤ |Λ L |.<br />
At the end of the proof, we will choose an optimal L.<br />
To estimate the right hand side <strong>in</strong> (6.6) from above we need an estimate of E 0 (H N Λ L<br />
)<br />
from below which will be provided by Temple’s <strong>in</strong>equality. As a test function ψ<br />
for Temple’s <strong>in</strong>equality we use the ground state of (H 0 ) N Λ L<br />
, namely<br />
ψ 0 (n) = 1<br />
|Λ L | 1 2<br />
for all n ∈ Λ L .
53<br />
In fact (H 0 ) N Λ L<br />
ψ 0 = 0. We have<br />
〈ψ 0 , HΛ N L<br />
ψ 0 〉 = 〈ψ 0 , V ω ψ 0 〉<br />
1 ∑<br />
=<br />
(2L + 1) d V ω (i) . (6.7)<br />
i∈Λ L<br />
Observe that this is an arithmetic mean of <strong>in</strong>dependent, identically distributed random<br />
variables. Hence, (6.7) converges <strong>to</strong> E(V ω (0)) > 0 almost surely.<br />
To apply Temple’s <strong>in</strong>equality, we would need<br />
1 ∑<br />
(2L + 1) d<br />
i∈Λ L<br />
V ω (i) < E 1 (H N Λ L<br />
)<br />
which is certa<strong>in</strong>ly wrong for large L s<strong>in</strong>ce E 1 (H N Λ L<br />
) → 0. We estimate<br />
E 1 (H N Λ L<br />
) ≥ E 1 ((H 0 ) N Λ L<br />
) ≥ cL −2 .<br />
The latter <strong>in</strong>equality can be obta<strong>in</strong>ed by direct calculation. Now we def<strong>in</strong>e<br />
V ω<br />
(L) (i) = m<strong>in</strong>{V ω (i),<br />
c<br />
3 L−2 } .<br />
For fixed L, the random variables V ω<br />
(L) are still <strong>in</strong>dependent and identically distributed,<br />
but their distribution depends on L. Moreover, if<br />
H (L) = (H 0 ) N Λ L<br />
+ V (L)<br />
ω<br />
then E 0 (H N Λ L<br />
) ≥ E 0 (H (L) ) by the m<strong>in</strong>-max pr<strong>in</strong>ciple (Theorem 3.1).<br />
We get<br />
〈ψ 0 , H (L) ψ 0 〉 =<br />
1 ∑<br />
(2L + 1) d<br />
i∈Λ L<br />
V (L)<br />
ω (i) ≤ c 3 L−2 (6.8)<br />
by def<strong>in</strong>ition of V ω<br />
(L) , consequently<br />
〈ψ 0 , H (L) ψ 0 〉 ≤ c 3 L−2 < E 1 ((H 0 ) N Λ L<br />
) ≤ E 1 (H (L) ) .<br />
Thus, we may use Temples <strong>in</strong>equality with ψ 0 and H (L) :
54<br />
E 0 (H N Λ L<br />
) ≥ E 0 (H (L) )<br />
≥ 〈ψ 0 , H (L) ψ 0 〉 − 〈ψ 0, (H (L) ) 2 ψ 0 〉<br />
cL −2 − 〈ψ 0 , H (L) ψ 0 〉<br />
≥<br />
≥<br />
1 ∑<br />
(2L + 1) d<br />
1 ∑<br />
(2L + 1) d<br />
V ω<br />
(L)<br />
i∈Λ L<br />
V ω<br />
(L)<br />
i∈Λ L<br />
≥ 1 1 ∑<br />
2 (2L + 1) d<br />
i∈Λ L<br />
V (L)<br />
(i) −<br />
(i) −<br />
Collect<strong>in</strong>g the estimates above, we arrive at<br />
1<br />
∑<br />
(2L+1) d i∈Λ L<br />
(V ω (i)) 2<br />
(c − c 3 )L−2<br />
1 ∑<br />
(2L + 1) d<br />
N(E) ≤ P ( E 0 (H N Λ L<br />
) < E ) ≤ P ( 1<br />
(2L + 1) d ∑<br />
V ω<br />
(L)<br />
i∈Λ L<br />
( c<br />
3 L−2<br />
)<br />
2<br />
3 cL−2<br />
ω (i) . (6.9)<br />
i∈Λ L<br />
V (L)<br />
ω < E 2<br />
)<br />
. (6.10)<br />
Now we choose L. We try <strong>to</strong> make the right hand side of (6.10) as small as possible.<br />
S<strong>in</strong>ce V ω<br />
(L) ≤ c 3 L−2 , the probability <strong>in</strong> (6.10) will be one if L is <strong>to</strong>o big.<br />
So we certa<strong>in</strong> want <strong>to</strong> choose L <strong>in</strong> such a way that c 3 L−2 > E 2 .<br />
Thus, a reasonable choice seems <strong>to</strong> be<br />
L := ⌊βE − 1 2 ⌋<br />
with some β small enough and ⌊x⌋ the largest <strong>in</strong>teger not exceed<strong>in</strong>g x.<br />
We s<strong>in</strong>gle out an estimate of the probability <strong>in</strong> (6.10)<br />
LEMMA 6.4. For L = ⌊βE − 1 2 ⌋ with β small and L large enough<br />
⎛<br />
⎞<br />
P ⎝ 1 ∑<br />
ω (i) < E ⎠ ≤ e −γ|Λ L|<br />
|Λ L |<br />
2<br />
with some γ > 0.<br />
Given the lemma, we proceed<br />
N(E) ≤ P<br />
i∈Λ L<br />
V (L)<br />
⎛<br />
⎝ 1<br />
|Λ L |<br />
≤ e −γ|Λ L|<br />
∑<br />
i∈Λ L<br />
V (L)<br />
= e −γ(2⌊βE− 1 2 ⌋ d +1)<br />
ω (i) < E 2<br />
≤ e −γ′ E − d2 . (6.11)<br />
⎞<br />
⎠
55<br />
This estimate is the desired upper bound on N(E).<br />
□<br />
To f<strong>in</strong>ish the proof of the upper bound, it rema<strong>in</strong>s <strong>to</strong> prove Lemma 6.4. This lemma<br />
is a typical large deviation estimate: By our choice of L we have E(V ω (L) ) > E 2<br />
if β is small enough; thus, we estimate the probability that an arithmetic mean of<br />
<strong>in</strong>dependent random variables deviates from its expectation value. What makes the<br />
problem somewhat nonstandard is the fact that the random variables V (L) depend<br />
on the parameter L, which is also implicit <strong>in</strong> E.<br />
PROOF (Lemma) :<br />
P ( 1 ∑<br />
V<br />
L<br />
|Λ L | ω (i) < E )<br />
2<br />
≤ P ( 1 ∑<br />
V<br />
L<br />
|Λ L | ω (i) < β2<br />
2 L−2)<br />
≤ P<br />
(#{ i | Vω L (i) < c )<br />
3 L−2 } ≥ (1 − 3β2<br />
c )|Λ L|<br />
. (6.12)<br />
Indeed, if less than (1 − 3β2 )|Λ L| of the V (i) are below c 3 L−2 than more than<br />
3β 2<br />
c<br />
c<br />
|Λ L| of them are at least c 3 L−2 (<strong>in</strong> fact equal <strong>to</strong>). In this case<br />
1 ∑<br />
V<br />
L<br />
|Λ L | ω (i)<br />
≥<br />
1 3β 2<br />
|Λ L | c |Λ L| c 3 L−2<br />
= β2<br />
2 L−2 .<br />
S<strong>in</strong>ce P (V (i)<br />
{<br />
> 0) > 0 there is a γ > 0 such that q := P (V (i) < γ) < 1.<br />
1 if Vi < γ,<br />
We set ξ i =<br />
0 otherwise.<br />
The random variables ξ i are <strong>in</strong>dependent and identically distributed, E(ξ i ) = q.<br />
Let us set r = 1 − 3β2<br />
c<br />
. By tak<strong>in</strong>g β small we can ensure that q < r < 1.<br />
Then, for L sufficient large<br />
(<br />
(6.12) ≤ P #{i | Vω L (i) < c )<br />
3 L−2 } ≥ r|Λ L |<br />
≤ P ( #{i | Vω L (i) < γ} ≥ r|Λ L | )<br />
≤ P( 1<br />
|Λ L |<br />
∑<br />
ξi ≥ r) . (6.13)<br />
Through our somewhat lengthy estimate above we f<strong>in</strong>ally arrived at the standard<br />
large deviations problem (6.13). To estimate the probability <strong>in</strong> (6.13) we use the<br />
<strong>in</strong>equality<br />
P (X > a) ≤ e −ta E(e tX ) for t ≥ 0 .
56<br />
Indeed<br />
P( X > a) =<br />
≤<br />
≤<br />
∫<br />
∫<br />
∫<br />
χ {X>a} (ω) d P(ω)<br />
e −ta e tX χ {X>a} (ω) d P(ω)<br />
e −ta e tX d P .<br />
We obta<strong>in</strong><br />
(6.13) ≤ e −|Λ L| t r · E( ∏<br />
i∈Λ L<br />
e tξ i<br />
)<br />
= e −|Λ L|(rt−ln E(e tξ 0 )) .<br />
Set f(t) = rt − ln E(e tξ 0<br />
). If we can choose t such that f(t) > 0, the result is<br />
proven. To see that this is possible, we compute<br />
f ′ (t) = r − E(ξ 0e tξ 0<br />
)<br />
E(e tξ 0 )<br />
So f ′ (0) = r − q > 0.<br />
S<strong>in</strong>ce f(0) = 0, there is a t > 0 with f(t) > 0.<br />
□<br />
Thus, we have shown<br />
lim E↘E0<br />
ln | ln N(E)|<br />
ln(E − E 0 ) ≤ −d 2 . (6.14)<br />
6.3. Lower bound.<br />
We proceed with the lower bound. By (5.50) we estimate<br />
N(E)<br />
≥<br />
≥<br />
1<br />
|Λ L | E ( N(HΛ D L<br />
, E) )<br />
1<br />
|Λ L | P ( E 0 (HΛ D L<br />
) < E ) . (6.15)<br />
As <strong>in</strong> the upper bound, the above estimate holds for any L.<br />
To proceed, we have <strong>to</strong> estimate E 0 (H D Λ L<br />
) from above.<br />
This is easily done via the m<strong>in</strong>-max pr<strong>in</strong>ciple (Theorem 3.1):<br />
E 0 (H D Λ L<br />
) ≤ 〈ψ, H D Λ L<br />
ψ〉<br />
= 〈ψ, (H 0 ) D Λ L<br />
ψ〉 + ∑<br />
i∈Λ L<br />
V ω (i) |ψ(i)| 2 (6.16)
for any ψ with ||ψ|| = 1. Now we try <strong>to</strong> f<strong>in</strong>d ψ which m<strong>in</strong>imizes the right hand<br />
side of (6.16). First we deal with the term<br />
〈ψ, (H 0 ) D Λ L<br />
ψ〉 . (6.17)<br />
S<strong>in</strong>ce (H 0 ) D Λ L<br />
adds a positive term <strong>to</strong> (H 0 ) N Λ L<br />
at the boundary, it seems desirable<br />
<strong>to</strong> choose ψ(n) = 0 for |n| = L. On the other hand, <strong>to</strong> keep (6.17) small we don’t<br />
want ψ <strong>to</strong> change <strong>to</strong>o abruptly.<br />
So, we choose<br />
57<br />
and<br />
ψ 1 (n) = L − || n|| ∞<br />
, n ∈ Λ L<br />
We have<br />
ψ(n) = 1<br />
||ψ 1 || ψ 1(n) .<br />
∑<br />
|ψ 1 (n)| 2 ≥ ∑<br />
n∈Λ L<br />
n∈Λ L2<br />
|ψ 1 (n)| 2 ≥ |Λ L<br />
2<br />
| ( L 2 )2 ≥ c L d+2<br />
and<br />
〈ψ 1 , (H 0 ) D Λ L<br />
ψ 1 〉 ≤<br />
∑<br />
∣ ψ 1 (n ′ ) − ψ 1 (n) ∣ ∣ 2<br />
≤<br />
(n,n ′ )∈Λ L<br />
|| n−n ′ || 1 =1<br />
∣ { (n, n ′ ) ∈ Λ L × Λ L ; || n − n ′ || 1 = 1} ∣<br />
≤ c 1 L d .<br />
Above, we used that |ψ 1 (n) − ψ 1 (n ′ )| ≤ 1 if || n − n ′ || 1 = 1.<br />
Collect<strong>in</strong>g these estimates, we obta<strong>in</strong><br />
E 0<br />
(<br />
(H 0 ) D Λ L<br />
)<br />
≤ 〈ψ 1, (H 0 ) D Λ L<br />
ψ 1 〉<br />
〈ψ 1 , ψ 1 〉<br />
≤ c 0 L −2 .<br />
The bounds of (6.15), (6.16) and (6.18) give<br />
⎛<br />
⎞<br />
1<br />
N(E) ≥<br />
|Λ L | P ⎝ ∑<br />
V ω (i) |ψ(i)| 2 < E − c 0 L −2 ⎠<br />
i∈Λ L<br />
≥<br />
(6.18)<br />
⎛<br />
⎞<br />
1<br />
|Λ L | P ⎝ c ∑<br />
2<br />
V ω (i) < E − c 0 L −2 ⎠ . (6.19)<br />
|Λ L |<br />
i∈Λ L/2
58<br />
In the last estimate, we used that for || i || ∞ ≤ L/2<br />
ψ(i) =<br />
1<br />
||ψ 1 || (L − || i|| ∞)<br />
≥ c L − || i|| ∞<br />
L d+2<br />
2<br />
≥ c L/2<br />
L d+2<br />
2<br />
= ˜c L −d/2 .<br />
The probability <strong>in</strong> (6.19) is aga<strong>in</strong> a large deviation probability. As above, the<br />
L−<strong>in</strong>dependence of the right hand side is nonstandard. We estimate (6.19) <strong>in</strong> a<br />
somewhat crude way by<br />
(6.19) ≥<br />
=<br />
1<br />
|Λ L | P ( For all i ∈ Λ L/2 , V ω (i) < 1 (E − c 0 L −2 ) )<br />
c 2<br />
1<br />
(<br />
|Λ L | P V ω (0) < 1 (E − c 0 L −2 |ΛL/2 |<br />
))<br />
. (6.20)<br />
c 2<br />
If we take L so large that c 0 L −2 < E 2 (i.e. L ∼ E−1/2 as for the upper bound), we<br />
obta<strong>in</strong><br />
(6.20) ≥ 1 (<br />
| Λ L | P c3 L<br />
0 [0, E/2)) d .<br />
Us<strong>in</strong>g assumption (6.3), we f<strong>in</strong>ally get<br />
N(E) ≥ c 4 L −d E c 3κL d<br />
= c 4 L −d e (ln E)c 3κL d .<br />
We rem<strong>in</strong>d the reader that κ is the exponent occurr<strong>in</strong>g <strong>in</strong> (6.3).<br />
So<br />
This gives the lower bound<br />
N(E) ≥ c ′ 4E d/2 e c′ 3 (ln E)E−d/2 .<br />
lim E↘E0<br />
ln | ln N(E)|<br />
ln(E − E 0 ) ≥ −d 2 .<br />
Notes and Remarks<br />
There are various approaches <strong>to</strong> Lifshitz tails by now. The first is through the<br />
Donsker-Varadhan theory of large deviations, see [38], [108] and [110]. Related<br />
results and further references can be found <strong>in</strong> [15]. For an alternative approach, see<br />
[132].<br />
The results conta<strong>in</strong>ed <strong>in</strong> these lecture and variants can be found for example <strong>in</strong><br />
[66], [69], [58], [121] and [129]. See also [72].
For an approach us<strong>in</strong>g periodic approximation see [81], [83], [84], [85], and references<br />
there<strong>in</strong>. In [102] the probabilistic and the spectral po<strong>in</strong>t of view were<br />
comb<strong>in</strong>ed.<br />
There are also results on other band edges than the bot<strong>to</strong>m of the spectrum, so<br />
called <strong>in</strong>ternal Lifshitz tails, [103], [122], [86], [87] and [107].<br />
Magnetic fields change the Lifshitz behavior drastically, see [20], [43], [139].<br />
For a recent survey about the density of states, see [67].<br />
59
61<br />
7. The spectrum and its physical <strong>in</strong>terpretation<br />
7.1. Generalized Eigenfunctions and the spectrum.<br />
In this section we explore the connection between generalized eigenfunctions of<br />
(discrete) Hamil<strong>to</strong>nians H and their spectra. A function f on Z d is called polynomially<br />
bounded if<br />
|f(n)| ≤ C ( 1 + || n|| ∞<br />
) k<br />
(7.1)<br />
for some constants k, C > 0. We say that λ is a generalized eigenvalue if there is<br />
a polynomially bounded solution ψ of the f<strong>in</strong>ite difference equation<br />
Hψ = λψ . (7.2)<br />
ψ is called a generalized eigenfunction. Note that we do not require ψ ∈ l 2 (Z d )!<br />
We denote the set of generalized eigenvalues of H by ε g (H).<br />
We say that the sets A, B ∈ B(R) agree up <strong>to</strong> a set of spectral measure zero if<br />
χ A\B (H) = χ B\A (H) = 0 where χ I (H) is the projection valued spectral measure<br />
associated with H (see Section 3.2).<br />
The goal of this section is <strong>to</strong> prove the follow<strong>in</strong>g theorem.<br />
THEOREM 7.1. The spectrum of a (discrete) Hamili<strong>to</strong>nian H agrees up <strong>to</strong> a set of<br />
spectral measure zero with the set ε g (H) of all generalized eigenvalues.<br />
As a corollary <strong>to</strong> the proof of Theorem 7.1 we obta<strong>in</strong> the follow<strong>in</strong>g result<br />
COROLLARY 7.2. <strong>An</strong>y generalized eigenvalue λ of H belongs <strong>to</strong> the spectrum<br />
σ(H), moreover<br />
σ(H) = ε g (H) (7.3)<br />
REMARK 7.3. The proof shows that <strong>in</strong> Theorem 7.1 as well as <strong>in</strong> Corollary 7.2<br />
the set ε g (H) can be replaced by the set of those generalized eigenvalues with a<br />
correspond<strong>in</strong>g generalized eigenfunction satisfy<strong>in</strong>g<br />
for some ε > 0.<br />
| ψ(n) | ≤ C ( 1 + || n|| ∞<br />
) d<br />
2 +ε (7.4)<br />
The proof of Theorem 7.1 and Corollary 7.2 we present now is quite close <strong>to</strong> [123],<br />
but the arguments simplify considerably <strong>in</strong> the discrete (l 2 -) case we consider here.<br />
For ∆ ⊂ R a Borel set, let µ(∆) = χ ∆ (H) be the projection valued measure<br />
associated with the self adjo<strong>in</strong>t opera<strong>to</strong>r H (see Section 3.2). Thus,<br />
with<br />
In the case of H = l 2 (Z d ), we set<br />
∫<br />
〈ϕ, Hψ〉 =<br />
λ dµ ϕ,ψ (λ)<br />
µ ϕ,ψ (∆) = 〈ϕ, µ(∆)ψ〉 .
62<br />
µ n,m (∆) = 〈δ n , µ(∆)δ m 〉 .<br />
If {α n } n∈Z d is a sequence of real numbers with α n > 0, ∑ α n = 1, we def<strong>in</strong>e<br />
ρ(∆) = ∑ n∈Z d α n µ n,n (∆) . (7.5)<br />
ρ is a f<strong>in</strong>ite positive Borel measure of <strong>to</strong>tal mass ρ(R) = 1. We call ρ a spectral<br />
measure (sometimes real valued spectral measure <strong>to</strong> dist<strong>in</strong>guish it from µ, the<br />
projection valued spectral measure). It is easy <strong>to</strong> see that<br />
ρ(∆) = 0 if and only if µ(∆) = 0 (7.6)<br />
Thus, A and B agree up <strong>to</strong> a set of spectral measure zero if ρ(A \ B) = 0 and<br />
ρ(B \ A) = 0. Moreover, the support of ρ is the spectrum of H. Although the<br />
spectral measure is not unique (many choices for the α n ), its measure class and its<br />
support are uniquely def<strong>in</strong>ed by (7.5).<br />
We are ready <strong>to</strong> prove one half of Theorem 7.1, namely<br />
PROPOSITION 7.4. Let ρ be a spectral measure for H = H 0 + V . Then, for ρ-<br />
almost all λ there exists a polynomially bounded solution of the difference equation<br />
Hψ = λψ .<br />
PROOF:<br />
By the Cauchy-Schwarz <strong>in</strong>equality, we have<br />
|µ n,m (∆)| ≤ µ n,n (∆) 1 2 µm,m (∆) 1 2<br />
Consequently, the µ n,m are absolutely cont<strong>in</strong>uous with respect <strong>to</strong> ρ, i.e.<br />
ρ(∆) = 0 =⇒ µ n,m (∆) = 0 . (7.7)<br />
Hence, the Radon-Nikodym theorem tells us that there exist measurable functions<br />
F n,m (densities) such that<br />
∫<br />
µ n,m (∆) = F n,m (λ) dρ(λ) . (7.8)<br />
∆<br />
The functions F n,m are def<strong>in</strong>ed up <strong>to</strong> sets of ρ-measure zero and, s<strong>in</strong>ce µ n,n ≥ 0<br />
the functions F n,n are non negative ρ-almost surely (ρ-a.s.). Moreover<br />
ρ(∆) = ∑ α n µ n,n<br />
= ∑ ∫<br />
α n F n,n (λ) dρ(λ)<br />
∆<br />
∫<br />
∑<br />
= αn F n,n (λ) dρ(λ) . (7.9)<br />
Hence, ∑ α n F n,n (λ) = 1 (ρ-a.s.). In particular<br />
∆
63<br />
It follows<br />
∫<br />
∣<br />
∆<br />
F n,m (λ) dρ(λ)<br />
∣ = |µ n,m (∆)|<br />
F n,n (λ) ≤ 1<br />
α n<br />
. (7.10)<br />
Thus<br />
≤ µ n,n (∆) 1 2 µm,m (∆) 1 2<br />
(∫<br />
= F n,n (λ) dρ(λ)<br />
∆<br />
) 1<br />
(∫ 2<br />
F m,m (λ) dρ(λ)<br />
∆<br />
) 1<br />
2<br />
≤ α − 1 2<br />
n α − 1 2<br />
m ρ(∆) . (7.11)<br />
|F n,m | ≤ α − 1 2<br />
n α − 1 2<br />
m . (7.12)<br />
Equation (7.8) implies that for any bounded measurable function f<br />
∫<br />
〈δ n , f(H)δ m 〉 = f(λ)F n,m (λ) dρ(λ) .<br />
In particular, for f(λ) = λ g(λ) (g of compact support)<br />
∫<br />
λ g(λ) F n,m (λ) dρ(λ)<br />
= 〈δ n , H g(H) δ m 〉<br />
= 〈H δ n , g(H) δ m 〉<br />
= ∑ (<br />
−〈δn+e , g(H) δ m 〉 ) + ( V (n) + 2d ) 〈δ n , g(H) δ m 〉<br />
|e|=1<br />
= ∑ ∫<br />
(<br />
−<br />
=<br />
|e|=1<br />
∫<br />
g(λ) F n+e,m (λ) dρ(λ) ) ∫<br />
+<br />
g(λ) (V (n) + 2d)F n,m (λ) dρ(λ)<br />
g(λ) H (n) F n,m (λ) dρ(λ) (7.13)<br />
where H (n) F n,m (λ) is the opera<strong>to</strong>r H applied <strong>to</strong> the function n ↦→ F n,m (λ). Thus,<br />
∫<br />
∫<br />
g(λ) λ F n,m (λ) dρ(λ) = g(λ) H (n) F n,m (λ) dρ(λ)<br />
for any bounded measurable function g with compact support.<br />
It follows that for ρ-almost all λ and for any fixed m ∈ Z d , the function ψ(n) =<br />
F n,m (λ) is a solution of Hψ = λψ. By (7.12) the function ψ satisfies<br />
| ψ(n)| ≤ C 0 α − 1 2<br />
n . (7.14)
64<br />
So far, the sequence α n has only <strong>to</strong> fulfill α n > 0 and ∑ α n = 1. Now, we choose<br />
α n = c (1 + || n|| ∞ ) −β for an arbitrary β > d, hence<br />
| ψ(n)| ≤ C (1 + || n|| ∞ ) d 2 +ε<br />
for an ε > 0.<br />
This proves the proposition as well as the estimate (7.4).<br />
□<br />
We turn <strong>to</strong> the proof of the opposite direction of Theorem 7.1. As usual, we equip<br />
Z d with the norm || n|| ∞ = max i=1,...,d |n i |. So Λ L = {|n| ≤ L} is a cube of side<br />
length 2L + 1.<br />
For a subset S of Z d we denote by ||ψ|| S the l 2 - norm of ψ over the set S.<br />
We beg<strong>in</strong> with a lemma:<br />
LEMMA 7.5. If ψ is polynomially bounded (≠ 0) and l is a positive <strong>in</strong>teger, then<br />
there is a sequence L n → ∞ such that<br />
‖ψ‖ ΛLn+l<br />
‖ψ‖ ΛLn<br />
→ 1 .<br />
PROOF: Suppose the assertion of the Lemma is wrong. Then there exists a > 1<br />
and L 0 such that for all L ≥ L 0<br />
‖ψ‖ ΛL+l ≥ a‖ψ‖ ΛL .<br />
So<br />
‖ψ‖ ≥ ΛL0 +lk ak ‖ψ‖ ΛL0 (7.15)<br />
but by the polynomial boundedness of ψ we have<br />
‖ψ‖ ≤ C ΛL0 +lk 1 (L 0 + l k) M ≤ C k M (7.16)<br />
for some C, M > 0 which contradicts (7.15).<br />
□<br />
We are now <strong>in</strong> position <strong>to</strong> prove the second half of Theorem 7.1.<br />
PROPOSITION 7.6. If the difference equation Hψ = λψ admits a polynomially<br />
bounded solution ψ, then λ belongs <strong>to</strong> the spectrum σ(H) of H.<br />
{<br />
ψ(n) for |n| ≤ L,<br />
PROOF: We set ψ L (n) =<br />
0 otherwise .<br />
Set ϕ L = 1<br />
||ψ L || ψ L. Then ψ L is ‘almost’ a solution of Hψ = λψ, more precisely<br />
(H − λ)ψ L (n) = 0<br />
as long as n ∉ S L := {m| L − 1 ≤ |m| ≤ L + 1}<br />
and
65<br />
‖(H − λ) ψ L || 2 ≤ || ψ|| 2 S L<br />
= ∑<br />
m∈S L<br />
| ψ(m)| 2<br />
By Lemma (7.5) there is a sequence L n → ∞ such that<br />
= || ψ|| 2 Λ L+1<br />
− ||ψ|| 2 Λ L−2<br />
. (7.17)<br />
so<br />
||ψ|| 2 Λ Ln+1<br />
||ψ|| 2 Λ Ln−2<br />
→ 1 ,<br />
||(H − λ)ϕ Ln || 2 ≤ ||ψ||2 Λ Ln+1<br />
− ||ψ|| 2 Λ Ln−2<br />
||ψ|| ΛLn−2<br />
→ 0 .<br />
Thus, ϕ Ln is a Weyl sequence for H and λ and λ ∈ σ(H).<br />
□<br />
PROOF (Corollary) :<br />
We have already seen <strong>in</strong> Proposition 7.6 that<br />
ε g (H) ⊂ σ(H) .<br />
S<strong>in</strong>ce σ(H) is closed, it follows<br />
By the Theorem 7.1, we know that<br />
ε g (H) ⊂ σ(H) .<br />
ρ ( ∁ ε g (H) ) = 0<br />
hence<br />
So,<br />
∁ ε g (H) ∩ σ(H) = ∁ ε g (H) ∩ supp ρ = ∅ .<br />
σ(H) ⊂ ε g (H) .<br />
□<br />
7.2. The measure theoretical decomposition of the spectrum.<br />
The spectrum gives the physically possible energies of the system described by the<br />
Hamil<strong>to</strong>nian H. Hence, if E /∈ σ(H), no (pure) state of the system can have energy<br />
E. It turns out that the f<strong>in</strong>e structure of the spectrum gives important <strong>in</strong>formation<br />
on the dynamical behavior of the system, more precisely on the long time behavior<br />
of the state ψ(t) = e −itH ψ 0 .<br />
To <strong>in</strong>vestigate this f<strong>in</strong>e structure we have <strong>to</strong> give a little background <strong>in</strong> measure<br />
theory. By the term bounded Borel measure (or bounded measure, for short) we
66<br />
mean <strong>in</strong> what follows a complex-valued σ-additive function ν on the Borel sets<br />
B(R) such that the <strong>to</strong>tal variation<br />
‖ν‖ = sup {<br />
N∑<br />
|ν(A i )|; A i ∈ B(R) pairwise disjo<strong>in</strong>t }<br />
i<br />
is f<strong>in</strong>ite. By a positive Borel measure we mean a non-negative σ-additive function<br />
m on the Borel sets such that m(A) is f<strong>in</strong>ite for any bounded Borel set A.<br />
A bounded Borel measure ν on R is called a pure po<strong>in</strong>t measure if ν is concentrated<br />
on a countable set, i.e. if there is a countable set A ∈ B(R) such that ν(R\A) = 0.<br />
The po<strong>in</strong>ts x i ∈ R with ν({x i }) ≠ 0 are called the a<strong>to</strong>ms of ν. A pure po<strong>in</strong>t<br />
measure ν can be written as ν = ∑ α i δ xi , where δ xi is the Dirac measure at the<br />
po<strong>in</strong>t x i and α i = ν({x i }).<br />
A measure ν is called cont<strong>in</strong>uous if ν has no a<strong>to</strong>ms, i.e. ν({x}) = 0 for all x ∈ R.<br />
A bounded measure ν is called absolutely cont<strong>in</strong>uous with respect <strong>to</strong> a positive<br />
measure m (<strong>in</strong> short ν ≪ m) if there is a measurable function ϕ ∈ L 1 (ν) such that<br />
ν = ϕ(m), i.e. ν(A) = ∫ A ϕ(x)dm(x).<br />
The Theorem of Radon-Nikodym asserts that ν is absolutely cont<strong>in</strong>uous with respect<br />
<strong>to</strong> m if (and only if) for any Borel set A, m(A) = 0 implies ν(A) = 0.<br />
By say<strong>in</strong>g ν is absolutely cont<strong>in</strong>uous we always mean ν is absolutely cont<strong>in</strong>uous<br />
with respect <strong>to</strong> Lebesgue measure L.<br />
A measure is called s<strong>in</strong>gular cont<strong>in</strong>uous if it is cont<strong>in</strong>uous and it lives on a set<br />
N of Lebesgue measure zero, i.e. ν({x}) = 0 for all x ∈ R, ν(R\N) = 0 and<br />
L(N) = 0.<br />
The Lebesgue-decomposition theorem tells us that any bounded Borel measure ν<br />
on R admits a unique decomposition<br />
ν = ν pp + ν sc + ν ac<br />
where ν pp is a pure po<strong>in</strong>t measure, ν sc a s<strong>in</strong>gular cont<strong>in</strong>uous measure and ν ac is<br />
absolutely cont<strong>in</strong>uous (with respect <strong>to</strong> Lebesgue measure). We call ν pp the pure<br />
po<strong>in</strong>t part of ν etc.<br />
Let H be a self adjo<strong>in</strong>t opera<strong>to</strong>r on a Hilbert space H with doma<strong>in</strong> D(H) and µ<br />
be the correspond<strong>in</strong>g projection valued spectral measure (see Section 3.2). So, for<br />
any Borel set A ⊂ R , µ(A) is a projection opera<strong>to</strong>r,<br />
is a complex valued measure and<br />
〈ϕ, µ(A)ψ〉 = µ ϕ,ψ (A)<br />
∫<br />
〈ϕ, Hψ〉 =<br />
λ dµ ϕ,ψ (λ) .<br />
We also set µ ϕ = µ ϕ,ϕ which is a positive measure. Note that
67<br />
| µ ψ,ϕ (A) | = | 〈ψ, µ(A)ϕ〉 |<br />
= | 〈µ(A)ψ, µ(A)ϕ〉 |<br />
≤ (〈µ(A)ψ, µ(A)ψ〉) 1 2 (〈µ(A)ϕ, µ(A)ϕ〉 )<br />
1<br />
2<br />
= µ ψ (A) 1 2 µϕ (A) 1 2 . (7.18)<br />
We def<strong>in</strong>e H pp = {ϕ ∈ H | µ ϕ is pure po<strong>in</strong>t} and analogously H sc and H ac .<br />
These sets are closed subspaces of H which are mutually orthogonal and<br />
H = H pp ⊕ H sc ⊕ H ac .<br />
The opera<strong>to</strong>r H maps each of these spaces <strong>in</strong><strong>to</strong> itself (see e.g. [117]). We set<br />
H pp = H | Hpp∩D(H), H sc = H | Hsc∩D(H) , H ac = H | Hac∩D(H) . We<br />
def<strong>in</strong>e the pure po<strong>in</strong>t spectrum σ pp (H) of H <strong>to</strong> be the spectrum σ(H pp ) of H pp ,<br />
analogously the s<strong>in</strong>gular cont<strong>in</strong>uous spectrum σ sc (H) of H <strong>to</strong> be σ(H sc ) and the<br />
absolutely cont<strong>in</strong>uous spectrum σ ac (H) <strong>to</strong> be σ(H ac ) . It is clear that<br />
σ(H) = σ pp (H) ∪ σ sc (H) ∪ σ ac (H)<br />
but this decomposition of the spectrum is not a disjo<strong>in</strong>t union <strong>in</strong> general.<br />
This measure theoretic decomposition of the spectrum is def<strong>in</strong>ed <strong>in</strong> a rather abstract<br />
way and we should ask: Is there any physical mean<strong>in</strong>g of the decomposition The<br />
answer is YES and will be given <strong>in</strong> the next section.<br />
7.3. Physical mean<strong>in</strong>g of the spectral decomposition.<br />
The measure theoretic decomposition of the Hilbert space and the spectrum may<br />
look more like a mathematical subtleness than like a physically relevant classification.<br />
In fact, <strong>in</strong> physics one is primarily <strong>in</strong>terested <strong>in</strong> long time behavior of wave<br />
packets. For example, one dist<strong>in</strong>guishes bound states and scatter<strong>in</strong>g states. It turns<br />
out, there is an <strong>in</strong>timate connection between the classification of states by their<br />
long time behavior and the measure theoretic decomposition of the spectrum. We<br />
explore this connection <strong>in</strong> the present section.<br />
The circle of results we present here was dubbed ‘RAGE-theorem’ <strong>in</strong> [30] after the<br />
pioneer<strong>in</strong>g works by Ruelle [120], Amre<strong>in</strong>, Georgescu [8] and Enss [42] on this<br />
<strong>to</strong>pic.<br />
If E is an eigenvalue of H (<strong>in</strong> the l 2 -sense) and ϕ a correspond<strong>in</strong>g eigenfunction,<br />
then the spectral measure µ has an a<strong>to</strong>m at E, and µ ϕ is a pure po<strong>in</strong>t measure<br />
concentrated at the po<strong>in</strong>t E. Thus, all eigenfunctions and the closed subspace<br />
generated by them belong <strong>to</strong> the pure po<strong>in</strong>t subspace H pp . The converse is also<br />
true, i.e. the space H pp is exactly the closure of the l<strong>in</strong>ear span of all eigenvec<strong>to</strong>rs.<br />
It follows that the set ε(H) of all eigenvalues of H is always conta<strong>in</strong>ed <strong>in</strong> the<br />
pure po<strong>in</strong>t spectrum σ pp (H) and that ε(H) is dense <strong>in</strong> σ pp (H). The set ε(H) is<br />
countable (as our Hilbert space is always assumed <strong>to</strong> be separable), it may have<br />
accumulation po<strong>in</strong>ts, <strong>in</strong> fact ε(H) may be dense <strong>in</strong> a whole <strong>in</strong>terval.
68<br />
Let us look at the time evolution of a function ψ ∈ H pp . To start with, suppose ψ<br />
is an eigenfunction of H with eigenvalue E. Then<br />
e −itH ψ = e −itE ψ<br />
so that |e −itH ψ(x)| 2 is <strong>in</strong>dependent of the time t. We may say that the particle, if<br />
start<strong>in</strong>g <strong>in</strong> an eigenstate, stays where it is for all t. It is easy <strong>to</strong> see that for general<br />
ψ <strong>in</strong> H pp the function e −itH ψ(x) is almost periodic <strong>in</strong> t. A particle <strong>in</strong> a state ψ <strong>in</strong><br />
H pp will stay <strong>in</strong>side a compact set with high probability for arbitrary long time, <strong>in</strong><br />
the follow<strong>in</strong>g sense:<br />
THEOREM 7.7. Let H be a self adjo<strong>in</strong>t opera<strong>to</strong>r on l 2 (Z d ), take ψ ∈ H pp and let<br />
Λ L denote a cube <strong>in</strong> Z d centered at the orig<strong>in</strong> with side length 2L + 1.<br />
Then<br />
⎛<br />
lim sup<br />
L→∞ t≥0<br />
⎞<br />
⎝ ∑<br />
| e −itH ψ(x) | 2 ⎠ = ‖ψ‖ 2 (7.19)<br />
x∈Λ L<br />
and<br />
lim sup<br />
L→∞ t≥0<br />
⎛<br />
⎞<br />
⎝ ∑<br />
| e −itH ψ(x) | 2 ⎠ = 0 . (7.20)<br />
x∉Λ L<br />
REMARK 7.8. Equations (7.19), (7.20) can be summarized <strong>in</strong> the follow<strong>in</strong>g way:<br />
Given any error bound ε > 0 there is a cube Λ L such that for arbitrary time t we<br />
f<strong>in</strong>d the particle <strong>in</strong>side Λ L with probability 1 − ε. In other words, the particle will<br />
not escape <strong>to</strong> <strong>in</strong>f<strong>in</strong>ity. Thus a state ψ ∈ H pp can be called a bound state .<br />
PROOF:<br />
S<strong>in</strong>ce e −itH is unitary, we have for all t<br />
|| ψ|| 2 = || e −itH ψ|| 2<br />
= ∑ x∈Λ<br />
| e −itH ψ (x) | 2 + ∑ x∉Λ<br />
| e −itH ψ (x) | 2 . (7.21)<br />
Consequently, (7.19) follows from (7.20).<br />
Above we saw that (7.20) is valid for eigenfunctions ψ. To prove it for other vec<strong>to</strong>rs<br />
<strong>in</strong> H pp , we <strong>in</strong>troduce the follow<strong>in</strong>g notation: By P L we denote the projection on<strong>to</strong><br />
∁Λ L . Then equation (7.20) claims that<br />
‖ P L e −itH ψ‖ → 0<br />
uniformly <strong>in</strong> t as L → ∞. If ψ is a (f<strong>in</strong>ite) l<strong>in</strong>ear comb<strong>in</strong>ation of eigenfunctions,<br />
say ψ = ∑ M<br />
m=1 α kψ k , Hψ k = E k ψ k , then
69<br />
‖ P L e −itH ψ‖ = ‖<br />
≤<br />
=<br />
M∑<br />
m=1<br />
M∑<br />
m=1<br />
M∑<br />
m=1<br />
α k P L e −itH ψ k ‖<br />
|α k | ‖ P L e −itH ψ k ‖ =<br />
M∑<br />
m=1<br />
|α k | ‖ P L e −itE k<br />
ψ k ‖<br />
|α k | ‖ P L ψ k ‖ . (7.22)<br />
By tak<strong>in</strong>g L large enough, each term <strong>in</strong> the sum above can be made smaller then<br />
( ∑M<br />
) −1<br />
m=1 |α k| ε .<br />
If now ψ is an arbitrary element of H pp , there is a l<strong>in</strong>ear comb<strong>in</strong>ation of eigenfunctions<br />
ψ (M) = ∑ M<br />
m=1 α kψ k such that ‖ψ − ψ (M) ‖ < ε. We conclude<br />
‖ P L e −itH ψ‖ ≤ ‖ P L e −itH ψ (M) ‖ + ‖ P L e −itH (ψ − ψ (M) )‖<br />
≤ ‖ P L e −itH ψ (M) ‖ + ‖ψ − ψ (M) ‖ . (7.23)<br />
By tak<strong>in</strong>g M large enough the second term of the right hand side can be made<br />
arbitrarily small. By choos<strong>in</strong>g L large, we can f<strong>in</strong>ally make the first term small as<br />
well.<br />
□<br />
We turn <strong>to</strong> the <strong>in</strong>terpretation of the cont<strong>in</strong>uous spectrum. Let us start with a vec<strong>to</strong>r<br />
ψ ∈ H ac . Then, by def<strong>in</strong>ition the spectral measure µ ψ is absolutely cont<strong>in</strong>uous.<br />
From estimate (7.18) we learn that µ ϕ,ψ is absolutely cont<strong>in</strong>uous for any ϕ ∈ H<br />
as well. It follows that the measure µ ϕ,ψ has a density h with respect <strong>to</strong> Lebesgue<br />
measure, <strong>in</strong> fact h ∈ L 1 . Hence, for any φ ∈ H and ψ ∈ H ac<br />
〈ϕ, e −itH ψ〉 =<br />
=<br />
∫<br />
∫<br />
e −itλ dµ ϕ,ψ (λ)<br />
e −itλ h(λ) d λ . (7.24)<br />
The latter expression is the Fourier transform of the (L 1 -)function h. Thus, by the<br />
Riemann-Lebesgue-Lemma (see e.g. [117]), it converges <strong>to</strong> 0 as t goes <strong>to</strong> <strong>in</strong>f<strong>in</strong>ity.<br />
We warn the reader that the decay of the Fourier transform of a measure does not<br />
imply that the measure is absolutely cont<strong>in</strong>uous. There are examples of s<strong>in</strong>gular<br />
cont<strong>in</strong>uous measures whose Fourier transforms decay.<br />
If the underly<strong>in</strong>g Hilbert space is l 2 (Z d ) we may choose ϕ = δ x for any x ∈ Z d ,<br />
then 〈ϕ, e −itH ψ〉 = e −itH ψ(x), thus we have immediately<br />
THEOREM 7.9. Let H be a self adjo<strong>in</strong>t opera<strong>to</strong>r on l 2 (Z d ), take ψ ∈ H ac and let<br />
Λ denote a f<strong>in</strong>ite subset of Z d . Then
70<br />
lim<br />
t→∞<br />
( ∑<br />
x∈Λ<br />
| e −itH ψ(x) | 2 )<br />
= 0 (7.25)<br />
or equivalently<br />
⎛<br />
⎞<br />
lim ⎝ ∑ | e −itH ψ(x) | 2 ⎠ = ‖ ψ ‖ 2 . (7.26)<br />
x∉Λ<br />
t→∞<br />
REMARK 7.10. As ψ ∈ H pp may be <strong>in</strong>terpreted as a particle stay<strong>in</strong>g (essentially)<br />
<strong>in</strong> a f<strong>in</strong>ite region for all time, a particle ψ ∈ H ac runs out <strong>to</strong> <strong>in</strong>f<strong>in</strong>ity as time evolves<br />
(and came out from <strong>in</strong>f<strong>in</strong>ity as t goes <strong>to</strong> −∞). So, <strong>in</strong> contrast <strong>to</strong> the bound states<br />
(ψ ∈ H pp ), we might call the states <strong>in</strong> H ac scatter<strong>in</strong>g states . Observe, however,<br />
that this term is used <strong>in</strong> scatter<strong>in</strong>g theory <strong>in</strong> a more restrictive sense.<br />
In the light of these results, states <strong>in</strong> the pure po<strong>in</strong>t subspace are <strong>in</strong>terpreted as<br />
bound states with low mobility. Consequently, electrons <strong>in</strong> such a state should<br />
not contribute <strong>to</strong> the electrical conductivity of the system. In contrast, states <strong>in</strong> the<br />
absolutely cont<strong>in</strong>uous subspace are highly mobile. They are the carrier of transport<br />
phenomena like conductivity.<br />
A (relatively) simple example of a quantum mechanical system is a Hydrogen<br />
a<strong>to</strong>m. After removal of the center of mass motion it consists of one particle mov<strong>in</strong>g<br />
. The spectrum of the correspond<strong>in</strong>g<br />
Schröd<strong>in</strong>ger opera<strong>to</strong>r consists of <strong>in</strong>f<strong>in</strong>itely many eigenvalues ( − C )<br />
n 2<br />
which accumulate at 0 and the <strong>in</strong>terval [0, ∞) represent<strong>in</strong>g the absolutely cont<strong>in</strong>uous<br />
spectrum. The eigenfunction correspond<strong>in</strong>g <strong>to</strong> the negative eigenvalues<br />
represent electrons <strong>in</strong> bound states, the orbitals. The states of the a.c.-spectrum<br />
correspond <strong>to</strong> electrons com<strong>in</strong>g from <strong>in</strong>f<strong>in</strong>ity be<strong>in</strong>g scattered at the nucleus and<br />
go<strong>in</strong>g off <strong>to</strong> <strong>in</strong>f<strong>in</strong>ity aga<strong>in</strong>.<br />
The Hydrogen a<strong>to</strong>m is typical for the classical picture of a quantum system: Above<br />
an energy threshold there is purely absolutely cont<strong>in</strong>uous spectrum due <strong>to</strong> scatter<strong>in</strong>g<br />
states, below the threshold there is a f<strong>in</strong>ite or countable set of eigenvalues accumulat<strong>in</strong>g<br />
at most at the threshold. For the harmonic oscilla<strong>to</strong>r there is a purely discrete<br />
spectrum, for periodic potentials the spectrum consists of bands with purely<br />
absolutely cont<strong>in</strong>uous spectrum. Until a few decades ago almost all physicists believed<br />
that all quantum systems belonged <strong>to</strong> one of the above spectral types.<br />
We have seen above that there may be pure po<strong>in</strong>t spectrum which is dense <strong>in</strong> a<br />
whole <strong>in</strong>terval and we will see this is <strong>in</strong> fact typically the case for random opera<strong>to</strong>rs.<br />
So far, we have not discussed the long time behavior for states <strong>in</strong> the s<strong>in</strong>gular cont<strong>in</strong>uous<br />
spectrum. S<strong>in</strong>gularly cont<strong>in</strong>uous spectrum seems <strong>to</strong> be particularly exotic<br />
and unnatural. In fact, one might tend <strong>to</strong> believe it is only a mathematical sophistication<br />
which never occurs <strong>in</strong> physics. This po<strong>in</strong>t of view is proved <strong>to</strong> be wrong.<br />
In fact, s<strong>in</strong>gularly cont<strong>in</strong>uous spectrum is typical for systems with aperiodic long<br />
range order, such as quasicrystals.<br />
The def<strong>in</strong>ition of s<strong>in</strong>gularly cont<strong>in</strong>uous measures is a quite <strong>in</strong>direct one. Indeed,<br />
we have not def<strong>in</strong>ed them by what they are but rather by what they are not. In other<br />
under the <strong>in</strong>fluence of a Coulomb potential V (x) = − Z<br />
|x|
words: S<strong>in</strong>gular cont<strong>in</strong>uous measures are those that rema<strong>in</strong> if we remove pure po<strong>in</strong>t<br />
and absolutely cont<strong>in</strong>uous measures. There is a characterization of cont<strong>in</strong>uous<br />
measures (i.e. those without a<strong>to</strong>ms) by their Fourier transform which goes back <strong>to</strong><br />
Wiener.<br />
THEOREM 7.11 (Wiener). Let µ be a bounded Borel measure on R and denote its<br />
Fourier transform by ˆµ(t) = ∫ e −itλ dµ(λ). Then<br />
71<br />
lim<br />
T →∞<br />
1<br />
T<br />
∫ T<br />
0<br />
| ˆµ(t)| 2 dt = ∑ x∈R<br />
| µ({x})| 2 .<br />
COROLLARY 7.12. µ is a cont<strong>in</strong>uous measure if and only if<br />
lim<br />
T →∞<br />
1<br />
T<br />
∫ T<br />
0<br />
| ˆµ(t)| 2 dt = 0 .<br />
PROOF (Theorem) :<br />
1<br />
T<br />
∫ T<br />
= 1 T<br />
∫<br />
=<br />
0<br />
∫ T<br />
R<br />
∫<br />
|ˆµ(t)| 2 dt<br />
0<br />
R<br />
(∫<br />
∫ )<br />
e −itλ dµ(λ) e itϱ d¯µ(ϱ) dt<br />
R<br />
R<br />
( ∫ 1 T<br />
)<br />
e it(ϱ−λ) dt d¯µ(ϱ) dµ(λ) . (7.27)<br />
T<br />
Here ¯µ denotes the complex conjugate of the measure µ. The functions<br />
0<br />
f T (ϱ, λ) = 1 T<br />
are bounded by one. Moreover for ϱ ≠ λ<br />
∫ T<br />
0<br />
e it(ϱ−λ) dt<br />
and<br />
f T (ϱ, λ) =<br />
1<br />
i(ϱ − λ)T (eiT (ϱ−λ) − 1) → 0 as T → ∞<br />
f T (ϱ, ϱ) = 1 .<br />
Thus, f T (ϱ, λ) → χ D (ϱ, λ) with D = {(x, y) | x = y}. By Lebesgue’s dom<strong>in</strong>ated<br />
convergence theorem, it follows that
72<br />
→<br />
=<br />
∫ T<br />
1<br />
|ˆµ(t)| 2 dt<br />
T 0<br />
∫ ∫<br />
χ D (ϱ, λ) d¯µ(ϱ)dµ(λ)<br />
∫<br />
¯µ({λ}) dµ(λ) = ∑ |µ({λ})| 2 . (7.28)<br />
This enables us <strong>to</strong> prove an analog of Theorem 7.7 and Theorem 7.9 for cont<strong>in</strong>uous<br />
measures. It says that states <strong>in</strong> the s<strong>in</strong>gularly cont<strong>in</strong>uous subspace represent<br />
particles which go off <strong>to</strong> <strong>in</strong>f<strong>in</strong>ity (at least) <strong>in</strong> the time average.<br />
THEOREM 7.13. Let H be a self adjo<strong>in</strong>t opera<strong>to</strong>r on l 2 (Z d ), take ψ ∈ H c and let<br />
Λ be a f<strong>in</strong>ite subset of Z d .<br />
Then<br />
⎛<br />
or equivalently<br />
lim<br />
T →∞<br />
lim<br />
T →∞<br />
1<br />
T<br />
1<br />
T<br />
∫ T<br />
0<br />
∫ T<br />
0<br />
⎞<br />
⎝ ∑ |e −itH ψ(x)| 2 ⎠ dt = ‖ ψ ‖ 2 (7.29)<br />
x∉Λ<br />
( )<br />
∑<br />
|e −itH ψ(x)| 2 dt = 0 . (7.30)<br />
x∈Λ<br />
PROOF: The equivalence of (7.29) and (7.30) follows from (7.21).<br />
We prove (7.30). Let ψ be <strong>in</strong> H c . From estimate (7.18), we learn that for any<br />
x ∈ Z d the measure µ δx,ψ is cont<strong>in</strong>uous. We have<br />
□<br />
∫ T<br />
1 ∑<br />
|e −itH ψ(x)| 2 dt = ∑ 1<br />
T 0<br />
T<br />
x∈Λ<br />
x∈Λ<br />
The latter term converges <strong>to</strong> 0 by Theorem (7.11).<br />
∫ T<br />
0<br />
|ˆµ δx,ψ| 2 dt .<br />
□<br />
We close this section with a result which allows us <strong>to</strong> express the projections on<strong>to</strong><br />
the pure po<strong>in</strong>t subspace and the absolutely cont<strong>in</strong>uous subspace as dynamical quantities.<br />
THEOREM 7.14. Let H be a self adjo<strong>in</strong>t opera<strong>to</strong>r on l 2 (Z d ), let P c and P pp be the<br />
orthogonal projection on<strong>to</strong> H c and H pp respectively, and let Λ L denote a cube <strong>in</strong><br />
Z d centered at the orig<strong>in</strong> with side length 2L + 1. Then, for any ψ ∈ l 2 (Z d )<br />
⎛<br />
⎞<br />
∫<br />
‖P c ψ‖ 2 = lim lim 1 T<br />
⎝ ∑<br />
| e −itH ψ(x) | 2 ⎠ dt (7.31)<br />
L→∞ T →∞ T 0<br />
x∉Λ L
73<br />
and<br />
‖P pp ψ‖ 2 = lim<br />
L→∞<br />
lim<br />
T →∞<br />
1<br />
T<br />
∫ T<br />
0<br />
⎛<br />
⎞<br />
⎝ ∑<br />
| e −itH ψ(x) | 2 ⎠ dt . (7.32)<br />
x∈Λ L<br />
PROOF:<br />
As <strong>in</strong> (7.21) we have<br />
|| P c ψ || 2 = 1 T<br />
− 1 T<br />
+ 1 T<br />
∫ T<br />
0<br />
∫ T<br />
0<br />
∫ T<br />
0<br />
( ∑<br />
x∉Λ L<br />
| e −itH ψ (x) | 2 ) dt<br />
( ∑<br />
| e −itH P pp ψ (x) | 2 ) dt<br />
x∉Λ L<br />
( ∑<br />
| e −itH P c ψ (x) | 2 ) dt . (7.33)<br />
x∈Λ L<br />
By Theorem 7.7 and Theorem 7.13 the second and the third term <strong>in</strong> (7.33) tend <strong>to</strong><br />
zero as T and (then) L go <strong>to</strong> <strong>in</strong>f<strong>in</strong>ity. This proves (7.31). Assertion (7.32) is proved<br />
<strong>in</strong> a similar way.<br />
□<br />
Notes and Remarks<br />
Most of the material <strong>in</strong> this chapter is based on [30], [123] and the lecture notes<br />
[134] by Gerald Teschl. Teschl’s excellent notes are only available on the <strong>in</strong>ternet.<br />
For further read<strong>in</strong>g we recommend [8], [13], [42], [113], [120] and [141].
75<br />
8.1. What physicists know.<br />
8. <strong>An</strong>derson localization<br />
S<strong>in</strong>ce the ground break<strong>in</strong>g work of P. <strong>An</strong>derson <strong>in</strong> the late fifties, physicists like<br />
Mott, Lifshitz, Thouless and many others have developed a fairly good knowledge<br />
about the measure theoretic nature of the spectrum of random Schröd<strong>in</strong>ger opera<strong>to</strong>rs,<br />
i.e. about the dynamical properties of wave packets.<br />
By Theorem 3.9 (see also Theorem 4.3) we know that the (almost surely non random)<br />
spectrum Σ of H ω is given by supp(P 0 ) + [0, 4d] where P 0 is the probability<br />
distribution of V ω (0). Thus if supp(P 0 ) consists of f<strong>in</strong>itely many po<strong>in</strong>ts or <strong>in</strong>tervals<br />
the spectrum Σ has a band structure <strong>in</strong> the sense that it is a union of (closed)<br />
<strong>in</strong>tervals.<br />
In the follow<strong>in</strong>g we report on the picture physicists developed about the measure<br />
theoretic structure of the spectrum of H ω . This picture is supported by conv<strong>in</strong>c<strong>in</strong>g<br />
physical arguments and is generally accepted among theoretical physicists. Only<br />
a part of it can be shown with mathematical rigor up <strong>to</strong> now. We will discuss this<br />
issue <strong>in</strong> the subsequent sections.<br />
There is a qualitative difference between one dimensional disordered systems<br />
(d = 1) and higher dimensional ones (d ≥ 3). For one dimensional (disordered)<br />
systems one expects that the whole spectrum is pure po<strong>in</strong>t. Thus, there is a complete<br />
system of eigenfunctions. The correspond<strong>in</strong>g (countably many) eigenvalues<br />
form a dense set <strong>in</strong> Σ (= ∪[a i , b i ]). The eigenfunctions decay exponentially at<br />
<strong>in</strong>f<strong>in</strong>ity. This phenomenon is called <strong>An</strong>derson localization or exponential localization.<br />
In the light of our discussion <strong>in</strong> section 7, we conclude that <strong>An</strong>derson<br />
localization corresponds <strong>to</strong> low mobility of the electrons <strong>in</strong> our system. Thus, one<br />
dimensional disordered systems (‘th<strong>in</strong> wires with impurities’) should have low or<br />
even vanish<strong>in</strong>g conductivity.<br />
In arbitrary dimension, an ordered quantum mechanical system should have purely<br />
absolutely cont<strong>in</strong>uous spectrum. This is known for periodic potentials <strong>in</strong> any dimension.<br />
Thus, <strong>in</strong> one dimension, an arbitrarily small disorder will change the<br />
<strong>to</strong>tal spectrum from absolutely cont<strong>in</strong>uous <strong>to</strong> pure po<strong>in</strong>t and hence a conduc<strong>to</strong>r <strong>to</strong><br />
an <strong>in</strong>sula<strong>to</strong>r. <strong>An</strong>derson localization <strong>in</strong> the one dimensional case can be proved with<br />
mathematical rigor for a huge class of disordered systems. We will not discuss the<br />
one dimensional case <strong>in</strong> detail <strong>in</strong> this paper.<br />
In dimension d ≥ 3 the physics of disordered systems is much richer (and consequently<br />
more complicated). As long as the randomness is not <strong>to</strong>o strong <strong>An</strong>derson<br />
localization occurs only near the band edges of the spectrum. Thus near any band<br />
edge a there is an <strong>in</strong>terval [a, a+δ] (resp. [a−δ, a]) of pure po<strong>in</strong>t spectrum and the<br />
correspond<strong>in</strong>g eigenfunctions are ‘exponentially localized’ <strong>in</strong> the sense that they<br />
decay exponentially fast at <strong>in</strong>f<strong>in</strong>ity.<br />
Well <strong>in</strong>side the bands, the spectrum is expected <strong>to</strong> be absolutely cont<strong>in</strong>uous at<br />
small disorder (d ≥ 3). S<strong>in</strong>ce the correspond<strong>in</strong>g (generalized) eigenfunctions are<br />
certa<strong>in</strong>ly not square <strong>in</strong>tegrable, one speaks of extended states or <strong>An</strong>derson delocalization<br />
<strong>in</strong> this regime. If the randomness of the system <strong>in</strong>creases the pure po<strong>in</strong>t
76<br />
spectrum will expand and the absolutely cont<strong>in</strong>uous part of the spectrum will shr<strong>in</strong>k<br />
correspond<strong>in</strong>gly. So, accord<strong>in</strong>g <strong>to</strong> physical <strong>in</strong>tuition, there is a phase transition<br />
from an <strong>in</strong>sulat<strong>in</strong>g phase <strong>to</strong> a conduct<strong>in</strong>g phase. A transition po<strong>in</strong>t between these<br />
phases is called a mobility edge.<br />
At a certa<strong>in</strong> degree of randomness, the a.c. spectrum should be ‘eaten up’ by the<br />
pure po<strong>in</strong>t spectrum. The physical implications of the above picture are that we<br />
expect an energy region for which the correspond<strong>in</strong>g states do not contribute <strong>to</strong> the<br />
conductance of the system (pure po<strong>in</strong>t spectrum) and an energy region correspond<strong>in</strong>g<br />
<strong>to</strong> states with good mobility which constitute the conductivity of the system<br />
(a.c. spectrum).<br />
In the above discussion we have deliberately avoided the case of space dimension<br />
d = 2. The situation <strong>in</strong> two dimensions was under debate <strong>in</strong> the theoretical physics<br />
community until a few years ago. At present, the general believe seems <strong>to</strong> be that<br />
we have complete <strong>An</strong>derson localization for d = 2 similar <strong>to</strong> the case d = 1.<br />
However, the pure po<strong>in</strong>t spectrum is expected <strong>to</strong> be less stable for d = 2, for<br />
example a magnetic field might be able <strong>to</strong> destroy it.<br />
8.2. What mathematicians prove.<br />
For more than 25 years, mathematicians have been work<strong>in</strong>g on random Schröd<strong>in</strong>ger<br />
opera<strong>to</strong>rs. Despite of this, the mathematically rigorous knowledge about these opera<strong>to</strong>rs<br />
is far from be<strong>in</strong>g complete.<br />
As mentioned above, the results on the one dimensional case are fairly satisfac<strong>to</strong>ry.<br />
One can prove <strong>An</strong>derson localization for all energies for a huge class of one<br />
dimensional random quantum mechanical systems.<br />
For quite a number of models <strong>in</strong> d ≥ 2 we also have proofs of <strong>An</strong>derson localization,<br />
even <strong>in</strong> the sense of dynamical localization (see Section 8.4), at low energies<br />
or high disorder. There are also results about localization at spectral edges (other<br />
than the bot<strong>to</strong>m of the spectrum).<br />
The model which is best unders<strong>to</strong>od <strong>in</strong> the cont<strong>in</strong>uous case is the alloy-type model<br />
with potential (2.5)<br />
V ω (x) = ∑ q i (ω)f(x − i) . (8.1)<br />
The q i are assumed <strong>to</strong> be <strong>in</strong>dependent with common distribution P 0 . Until very<br />
recently, all known localization proof (for d ≥ 2) required some k<strong>in</strong>d of regularity<br />
of the probability measure P 0 , for example the existence of a bounded density<br />
with respect <strong>to</strong> Lebesgue measure. In any case, these assumptions exclude the case<br />
when P 0 is concentrated <strong>in</strong> f<strong>in</strong>itely many po<strong>in</strong>ts. From a physical po<strong>in</strong>t of view<br />
such measures with a f<strong>in</strong>ite support are pretty natural. They model a random alloy<br />
with f<strong>in</strong>itely many constituents. A few years ago, Bourga<strong>in</strong> and Kenig [19] proved<br />
localization for the Bernoulli alloy type model, i.e. a potential as <strong>in</strong> (8.1) with P 0<br />
concentrated on {0, 1}.<br />
Their proof works <strong>in</strong> the cont<strong>in</strong>uous case, but it does not for the (discrete) <strong>An</strong>derson<br />
model. In the cont<strong>in</strong>uous case Bourga<strong>in</strong> and Kenig strongly use that eigenfunctions
of a Schröd<strong>in</strong>ger opera<strong>to</strong>r on R d can not decay faster than a certa<strong>in</strong> exponential<br />
bound. This is a strong quantitative version of the unique cont<strong>in</strong>uation theorem<br />
which says that a solution of the Schröd<strong>in</strong>ger equation which is zero on an open<br />
set vanishes everywhere.<br />
Such a unique cont<strong>in</strong>uation theorem is wrong on the lattice, so a fortiori the lower<br />
bound on eigenfunctions is not valid on Z d . This is the ma<strong>in</strong> reason why the proof<br />
by Bourga<strong>in</strong>-Kenig does not extend <strong>to</strong> the discrete case.<br />
Us<strong>in</strong>g ideas from Bourga<strong>in</strong>-Kenig [19], Germ<strong>in</strong>et, Hislop and Kle<strong>in</strong> [48] proved<br />
<strong>An</strong>derson localization for the Poisson model (2.8). Until their paper noth<strong>in</strong>g was<br />
known about <strong>An</strong>derson localization for the Poisson model <strong>in</strong> dimension d ≥ 2.<br />
(For d = 1 see [131]).<br />
It is certa<strong>in</strong>ly fair <strong>to</strong> say that by now mathematicians know quite a bit about <strong>An</strong>derson<br />
localization, i.e. about the <strong>in</strong>sulat<strong>in</strong>g phase.<br />
The contrary is true for <strong>An</strong>derson delocalization. There is no proof of existence of<br />
absolutely cont<strong>in</strong>uous spectrum for any of the models we have discussed so far. In<br />
particular it is not known whether there is a conduct<strong>in</strong>g phase or a mobility edge at<br />
all.<br />
Existence of absolutely cont<strong>in</strong>uous spectrum is known, however, for the so called<br />
Bethe lattice (or Cayley tree). This is a graph (”lattice”) without loops (hence a<br />
tree) with a fixed number of edges at every site. One considers the graph Laplacian<br />
on the Bethe lattice, which is analogously def<strong>in</strong>ed <strong>to</strong> the Laplacian on the graph<br />
Z d (see [75], [76], [77]) and an <strong>in</strong>dependent identically distributed potential on the<br />
sites of the graph.<br />
There are also ‘<strong>to</strong>y’-models similar <strong>to</strong> the <strong>An</strong>derson model but with non identically<br />
distributed V ω (i) which are more and more diluted (or ‘weak’) as || i|| ∞ becomes<br />
large. For these models, the mobility can be determ<strong>in</strong>ed. (see [61], [60] and [54]).<br />
77<br />
8.3. Localization results.<br />
We state the localization result we are go<strong>in</strong>g <strong>to</strong> prove <strong>in</strong> the next chapters. For<br />
convenience, we repeat our assumptions. They are stronger than necessary but<br />
allow for an easier, we hope more transparent, proof.<br />
Assumptions:<br />
(1) H 0 is the f<strong>in</strong>ite difference Laplacian on l 2 (Z d ).<br />
(2) V ω (i), i ∈ Z d are <strong>in</strong>dependent random variables with a common distribution<br />
P 0 .<br />
(3) P 0 has a bounded density g, i.e. P(V ω (i) ∈ A) = P 0 (A) = ∫ A g(λ)dλ<br />
and || g || ∞ < ∞.<br />
(4) supp P 0 is compact.<br />
DEFINITION 8.1. We say that the random opera<strong>to</strong>r H ω exhibits spectral localization<br />
<strong>in</strong> an energy <strong>in</strong>terval I (with I ∩ σ(H ω ) ≠ ∅) if for P-almost all ω<br />
σ c (H ω ) ∩ I = ∅ . (8.2)
78<br />
We will show spectral localization for low energies and for strong disorder. To<br />
measure the degree of disorder of P 0 = g dλ, we <strong>in</strong>troduce the ‘disorder parameter’<br />
δ(g) := ||g|| −1<br />
∞ . If δ(g) is large, i.e. ||g|| ∞ is small, then the probability density<br />
g (recall ∫ g = 1) is rather extended. So one may, <strong>in</strong> deed, say that δ(g) large is<br />
an <strong>in</strong>dica<strong>to</strong>r for large disorder. (If δ(g) is small then g might be concentrated near<br />
a small number of po<strong>in</strong>ts. This, however, is not a conv<strong>in</strong>c<strong>in</strong>g <strong>in</strong>dica<strong>to</strong>r of small<br />
disorder.) Let us denote by E 0 the bot<strong>to</strong>m of the (almost surely constant) spectrum<br />
of H ω = H 0 + V ω .<br />
In the follow<strong>in</strong>g chapters we will prove:<br />
THEOREM 8.2. There exists E 1 > E 0 = <strong>in</strong>f(σ(H ω )) such that the spectrum of<br />
H ω exhibits spectral decomposition <strong>in</strong> the <strong>in</strong>terval I = [E 0 , E 1 ].<br />
In particular, the spectrum <strong>in</strong>side I is pure po<strong>in</strong>t almost surely and the correspond<strong>in</strong>g<br />
eigenfunctions decay exponentially.<br />
THEOREM 8.3. For any <strong>in</strong>terval I ≠ ∅, there is a δ 0 such that for any δ(g) ≥ δ 0<br />
the opera<strong>to</strong>r of H ω exhibits spectral localization <strong>in</strong> I.<br />
The spectrum <strong>in</strong>side I is pure po<strong>in</strong>t almost surely and the correspond<strong>in</strong>g eigenfunctions<br />
decay exponentially.<br />
8.4. Further Results.<br />
As we discussed <strong>in</strong> the previous chapter, physicists are not primarily <strong>in</strong>terested <strong>in</strong><br />
spectral properties of random Hamil<strong>to</strong>nians but rather <strong>in</strong> dynamical properties, i.e.<br />
<strong>in</strong> the longtime behavior of e −itHω . Consequently <strong>An</strong>derson localization should<br />
have dynamical consequences, as we might expect from the considerations <strong>in</strong> section<br />
7.3.<br />
It seems reasonable <strong>to</strong> expect that the follow<strong>in</strong>g property holds <strong>in</strong> the localization<br />
regime.<br />
DEFINITION 8.4. We say that the random opera<strong>to</strong>r H ω exhibits dynamical localization<br />
<strong>in</strong> an energy <strong>in</strong>terval I (with I ∩ σ(H ω ) ≠ ∅) if for all ϕ <strong>in</strong> the Hilbert<br />
space and all p ≥ 0<br />
for P-almost all ω.<br />
sup<br />
t∈R<br />
|| |X| p e −itHω χ I (H ω ) ϕ || < ∞ (8.3)<br />
Above, χ I (H ω ) denotes the spectral projection for H ω on<strong>to</strong> the <strong>in</strong>terval I (see Section<br />
3.2) and |X| is the multiplication opera<strong>to</strong>r def<strong>in</strong>ed by |X| ψ(n) = || n|| ∞ ψ(n).<br />
Intuitively, dynamical localization tells us that the particle is concentrated near the<br />
orig<strong>in</strong> uniformly for all times. We will not prove dynamical localization here. We<br />
refer <strong>to</strong> the references given <strong>in</strong> the notes and <strong>in</strong> particular <strong>to</strong> the review [78].<br />
We turn <strong>to</strong> the question of the relation between spectral and dynamical localization.<br />
THEOREM 8.5. Dynamical localization implies spectral localization.
PROOF:<br />
From Theorem 7.14 we know<br />
⎛<br />
⎞<br />
∫<br />
‖P c ψ‖ 2 = lim lim 1 T<br />
⎝ ∑<br />
| e −itH ψ(j) | 2 ⎠ dt . (8.4)<br />
L→∞ T →∞ T 0<br />
j ∉Λ L<br />
For ψ = P I (H ω ) ϕ, we have<br />
∑<br />
| e −itH ψ(j) | 2 = ∑ 1<br />
j ∉Λ L<br />
|| j|| 2p ∣ |X| p e −itH ∣<br />
ψ(j)<br />
|| j || ∞<br />
∞>L<br />
≤ ∣ ∣ |X| p e −itHω ψ ∣ ∣ ∑<br />
1<br />
|| j|| 2p ∞<br />
By (8.3) we have that<br />
lim<br />
T →∞<br />
1<br />
T<br />
Thus, for p large enough,<br />
∫ T<br />
0<br />
‖P c ψ‖ 2 ≤ C lim<br />
L→∞<br />
|| j || ∞>L<br />
∣ 2<br />
79<br />
(8.5)<br />
∣ ∣ ∣ |X| p e −itHω ψ ∣ ∣ ∣ ∣ dt ≤ C < ∞ . (8.6)<br />
∑<br />
|| j || ∞>L<br />
1<br />
|| j|| 2p ∞<br />
Hence, there is only pure po<strong>in</strong>t spectrum <strong>in</strong>side the <strong>in</strong>terval I.<br />
= 0 . (8.7)<br />
It turns out that the converse is not true, <strong>in</strong> general. There are examples of opera<strong>to</strong>rs<br />
with pure po<strong>in</strong>t spectrum without dynamical localization [35].<br />
□<br />
Notes and Remarks<br />
For an overview on the physics of <strong>An</strong>derson localization / delocalization we refer<br />
<strong>to</strong> the papers [9], [98] and [135], [136]. For the mathematical aspects we refer <strong>to</strong><br />
[23], [112] and [128].<br />
In this lecture notes we have <strong>to</strong> omit many important results about the one dimensional<br />
case. We just mention a few of the most important papers about one<br />
dimensional localization here: [52], [104], [111], [88], as well as [22], [21], [31].<br />
In the multidimensional case there exist two quite different approaches <strong>to</strong> localization.<br />
The first (<strong>in</strong> chronological order) is the multiscale analysis based on the<br />
fundamental paper [47]. This is the method we are go<strong>in</strong>g <strong>to</strong> present <strong>in</strong> the follow<strong>in</strong>g<br />
chapters. For further references see the literature cited there.<br />
The second method, the method of fractional moments, is also called the Aizenman-<br />
Molchanov method after the basic paper [4]. At least for the lattice case, this<br />
method is <strong>in</strong> many ways easier than the multiscale analysis. Moreover, it gives a<br />
number of additional results. On the other hand its adaptation <strong>to</strong> the cont<strong>in</strong>uous<br />
case is rather <strong>in</strong>volved. We refer <strong>to</strong> [1], [53], [3], [5], [2] for further developments.<br />
We will not discuss this method here due <strong>to</strong> the lack of space and time.
80<br />
It was realized by Mart<strong>in</strong>elli and Scoppolla [101] that the result of multiscale analysis<br />
implies absence of a.c. spectrum. The first proofs of spectral localization<br />
were given <strong>in</strong>dependently <strong>in</strong> [46] (see also [40]), [37] and [126] . The latter papers<br />
develop the method of spectral averag<strong>in</strong>g which goes partly back <strong>to</strong> [89].<br />
The result, that dynamical localization implies spectral localization, was proved<br />
<strong>in</strong> [94] (see also [30]). These authors also prove dynamical localization for the<br />
one dimensional case. <strong>An</strong> example with spectral localization which fails <strong>to</strong> exhibit<br />
dynamical localization was given <strong>in</strong> [35].<br />
The first proof of dynamical localization for the multidimensional <strong>An</strong>derson model<br />
was given by Aizenman [1]. This paper uses and extends the Aizenman-Molchanov<br />
method of fractional moments <strong>to</strong> prove localization. De Bièvre and Germ<strong>in</strong>et [14]<br />
proved dynamical localization both for the <strong>An</strong>derson model and for alloy-type<br />
models (<strong>in</strong> the cont<strong>in</strong>uum). Their proof is based on multiscale ideas. Damanik<br />
and S<strong>to</strong>llmann [32] proved that the multiscale analysis actually implies dynamical<br />
localization. They proved a version of dynamical localization (strong dynamical<br />
localization) which is stronger than ours.<br />
There are various even stronger versions of dynamical localization, we just mention<br />
strong Hilbert-Schmidt dynamical localization which was proven by Germ<strong>in</strong>et and<br />
Kle<strong>in</strong> [49]. We refer <strong>to</strong> the survey [78] by Abel Kle<strong>in</strong> for this k<strong>in</strong>d of questions.<br />
In theoretical physics, the theory of conductivity goes much beyond a characterization<br />
of the spectral type of the Hamil<strong>to</strong>nian. One of the ma<strong>in</strong> <strong>to</strong>pics is the l<strong>in</strong>ear<br />
response theory and the Kubo-formula. This approach is <strong>in</strong>vestigated from a mathematical<br />
po<strong>in</strong>t of view <strong>in</strong> [3], [17], [79] (see also [74]).<br />
For delocalization on the Bethe lattice see: [75], [76], [77]. See also [7], [6] and<br />
[45] for new proofs and further developments.<br />
Delocalization for potentials with randomness decay<strong>in</strong>g at <strong>in</strong>f<strong>in</strong>ity was <strong>in</strong>vestigated<br />
<strong>in</strong> [92], [93], [60], [18], [118]. A localization / delocalization transition was proved<br />
for such potentials <strong>in</strong> [61], [54].<br />
Dynamical delocalization was shown for a random dimer model <strong>in</strong> [56] and for a<br />
random Landau Hamil<strong>to</strong>nian <strong>in</strong> [51]. Dynamical delocalization means that dynamical<br />
localization is violated <strong>in</strong> some sense. It does not imply delocalization <strong>in</strong> the<br />
sense of a.c. spectrum. Moreover, <strong>in</strong> the above cited papers dynamical delocalization<br />
is only shown at special energies of Lebesgue measure zero.
81<br />
9. The Green’s function and the spectrum<br />
9.1. Generalized eigenfunctions and the decay of the Green’s function.<br />
Here we start <strong>to</strong> prove <strong>An</strong>derson localization via the multiscale method. The proof<br />
will require the whole rest of this text. For the reader who might get lost while<br />
try<strong>in</strong>g <strong>to</strong> understand the proof, we provided a roadmap through these chapters <strong>in</strong><br />
chapter 12.<br />
We beg<strong>in</strong> our discuss of multiscale analysis. This method is used <strong>to</strong> show exponential<br />
decay of Green’s functions. In this section we <strong>in</strong>vestigate some of the<br />
consequences of that estimate on the spectral properties of H ω . The multiscale<br />
estimates are discussed <strong>in</strong> the next chapter.<br />
Let us start by def<strong>in</strong><strong>in</strong>g what we mean by exponential decay of Green’s functions .<br />
We recall some of the notations <strong>in</strong>troduced <strong>in</strong> previous chapters. Λ L (n) is the cube<br />
of side length (2L + 1) centered at n ∈ Z d (see (3.5)), and Λ L denotes a cube<br />
around the orig<strong>in</strong>. || m|| ∞ = sup i=1,...,d |m i |.<br />
The <strong>in</strong>ner boundary ∂ − Λ L (n) of Λ L (n) consists of the outermost layer of lattice<br />
po<strong>in</strong>ts <strong>in</strong> Λ L (n), namely (see 5.28)<br />
∂ − Λ L (n) = {m ∈ Z d | m ∈ Λ L (n), ∃ m ′ ∉ Λ L (n) (m, m ′ ) ∈ ∂Λ L (n)}<br />
= {m ∈ Z d | || m − n|| ∞ = L} . (9.1)<br />
Similarly, the outer boundary of Λ L (n) is def<strong>in</strong>ed by<br />
∂ + Λ L (n) = {m ∈ Z d | m ∉ Λ L (n), ∃ m ′ ∈ Λ L (n) (m, m ′ ) ∈ ∂Λ L (n)}<br />
= {m ∈ Z d | || m − n|| ∞ = L + 1} . (9.2)<br />
For A ⊂ Z d we denote the number of lattice po<strong>in</strong>ts <strong>in</strong>side A by |A|. So, |Λ L | =<br />
(2L+1) d and |∂ − Λ L | = 2d (2L) d−1 . By A m ↗ Z d we mean: A m ⊂ A m+1 ⊂ Z d<br />
and ⋃ A m = Z d .<br />
The Green’s function G Λ E (n, m) is the kernel of the resolvent of H Λ given by<br />
G Λ E (n, m) = (H Λ − E) −1 (n, m) = 〈 δ n , (H Λ − E) −1 δ m 〉 . (9.3)<br />
DEFINITION 9.1.<br />
(1) We will say that the Green’s functions G Λ L(n 0 )<br />
E<br />
(n, m) for energy E and<br />
potential V decays exponentially on Λ L (n 0 ) with rate γ (γ > 0) if E is<br />
not an eigenvalue for H ΛL (n 0 ) = (H 0 + V ) ΛL (n 0 ) and<br />
|G Λ L(n 0 )<br />
E<br />
(n, m)| = |(H ΛL − E) −1 (n, m) | ≤ e −γL (9.4)<br />
for all n ∈ Λ L 1/2(n 0 ) and all m ∈ ∂ − Λ L (n 0 ).<br />
(2) If the Green’s function G Λ L(n 0 )<br />
E<br />
decays exponentially with rate γ > 0 we<br />
call the cube Λ L (n 0 ) (γ, E)-good for V .
82<br />
(3) We call an energy E γ-good (for V ω ) if the there is a sequence of<br />
cubes Λ lm ↗ Z d such that all Λ lm are (γ, E)-good. (Note, that γ is<br />
<strong>in</strong>dependent of Λ lm !)<br />
Note that, by def<strong>in</strong>ition, E ∉ σ(H ΛL (n 0 )) if Λ L (n 0 ) is (γ, E)-good.<br />
The behavior of the Green’s function has important consequences for the behavior<br />
of (generalized) eigenfunctions. Suppose that the function ψ is a solution of the<br />
difference equation<br />
Hψ = Eψ . (9.5)<br />
Then (see equations (5.30) and (5.31))<br />
0 = (H − E)ψ = (H Λ ⊕ H ∁Λ + Γ Λ − E)ψ , (9.6)<br />
hence<br />
((H Λ ⊕ H ∁Λ ) − E)ψ = −Γ Λ ψ . (9.7)<br />
So, for any n 0 ∈ Λ we have<br />
(H Λ − E)ψ(n 0 ) = (−Γ Λ ψ)(n 0 ) . (9.8)<br />
Suppose that E is not an eigenvalue of H Λ , then (n 0 ∈ Λ)<br />
So<br />
ψ(n 0 ) = −[(H Λ − E) −1 Γ Λ ψ](n 0 ) . (9.9)<br />
ψ(n 0 ) = −<br />
∑<br />
(k,m)∈∂Λ<br />
k∈∂ − Λ,m∈∂ + Λ<br />
This enables us <strong>to</strong> prove a crucial observation.<br />
G Λ E(n 0 , k)ψ(m) . (9.10)<br />
THEOREM 9.2. If E is γ-good for V then E is not a generalized eigenvalue of<br />
H = H 0 + V .<br />
PROOF: Suppose ψ is a polynomially bounded eigenfunction of H with (generalized)<br />
eigenvalue E, hence<br />
Hψ = E ψ and |ψ(m)| ≤ c |m| r for m ≠ 0 .<br />
Take any n ∈ Z d , then n ∈ Λ 1/2 L<br />
(n 0 ) for k large enough. Thus by (9.10)<br />
k
83<br />
|ψ(n)|<br />
≤<br />
≤<br />
∣<br />
∑<br />
(m ′ ,m)∈ ∂Λ Lk<br />
m ′ ∈ Λ Lk<br />
G Λ L k<br />
E<br />
(n, m′ )ψ(m)<br />
c 1 L d−1<br />
k<br />
e −γL k<br />
sup<br />
m∈∂ + Λ Lk<br />
|ψ(m)| (9.11)<br />
≤ c 2 L d−1+r<br />
k<br />
e −γL k<br />
(9.12)<br />
→ 0 as k → ∞ . (9.13)<br />
∣<br />
Hence ψ ≡ 0. Consequently, there are no non zero polynomially bounded eigensolutions.<br />
□<br />
There are two immediate yet remarkable consequences of Theorem (9.2).<br />
COROLLARY 9.3. If every E ∈ [E 1 , E 2 ] is γ-good for V then<br />
σ(H 0 + V ) ∩ (E 1 , E 2 ) = ∅.<br />
COROLLARY 9.4. If Lebesgue-almost all E ∈ [E 1 , E 2 ] are γ-good for V then<br />
σ ac (H 0 + V ) ∩ (E 1 , E 2 ) = ∅ .<br />
PROOF: The assumption of Corollary 9.3 implies by Theorem 9.2 that there are<br />
no generalized eigenvalues <strong>in</strong> (E 1 , E 2 ). By Theorem 7.1 (or Proposition 7.4) it<br />
follows that there is no spectrum there.<br />
If there is any absolutely cont<strong>in</strong>uous spectrum <strong>in</strong> (E 1 , E 2 ) the spectral measure restricted<br />
<strong>to</strong> that <strong>in</strong>terval must have an absolutely cont<strong>in</strong>uous component. Hence, by<br />
Theorem 7.1, there must be a set of generalized eigenvalues of positive Lebesgue<br />
measure. However, this is not possible by the assumption of Corollary 9.4 and<br />
Theorem 9.2.<br />
□<br />
9.2. From multiscale analysis <strong>to</strong> absence of a.c. spectrum.<br />
The results of the previous section <strong>in</strong>dicate a close relation between the existence<br />
of (γ, E)-good cubes and the spectrum of the (discrete) Schröd<strong>in</strong>ger opera<strong>to</strong>r. The<br />
follow<strong>in</strong>g theorem gives first h<strong>in</strong>ts <strong>to</strong> a probabilistic analysis of this connection.<br />
THEOREM 9.5. If there is a sequence R k → ∞ of <strong>in</strong>tegers such that for every k,<br />
every E ∈ I = [E 1 , E 2 ] and a constant γ > 0<br />
then with probability one<br />
P ( Λ Rk is not (γ, E)-good ) → 0 (9.14)<br />
σ ac (H ω ) ∩ (E 1 , E 2 ) = ∅ . (9.15)
84<br />
PROOF: Set p k = P ( Λ Rk is not (γ, E)-good ). By pass<strong>in</strong>g <strong>to</strong> a subsequence, if<br />
necessary, we may assume that the R k are <strong>in</strong>creas<strong>in</strong>g and that ∑ p k < ∞.<br />
Consequently, from the Borel-Cantelli-Lemma (see Theorem 3.6)) we learn that<br />
with probability one, there is a k 0 such that all Λ Rk are (γ, E)−good for k ≥ k 0 .<br />
Hence for P−almost every ω any given E ∈ [E 1 , E 2 ] is γ−good.<br />
We set<br />
N = { (E, ω) ∈ [E 1 , E 2 ] × Ω ∣ }<br />
E is not γ−good for V ω (9.16)<br />
N E = { ω ∈ Ω ∣ ∣ E is not γ−good for V ω } (9.17)<br />
N ω = { E ∈ [E 1 , E 2 ] ∣ ∣ E is not γ−good for V ω<br />
}<br />
. (9.18)<br />
Above we proved P(N E ) = 0 for any E ∈ [E 1 , E 2 ].<br />
Denot<strong>in</strong>g the Lebesgue measure on R by λ we have by Fub<strong>in</strong>i’s theorem<br />
λ ⊗ P (N ) =<br />
=<br />
∫ E2<br />
∫<br />
E 1<br />
P(N E ) dλ(E)<br />
λ(N ω ) d P(ω) .<br />
S<strong>in</strong>ce P(N E ) = 0 for all E ∈ [E 1 , E 2 ] we conclude that<br />
∫ E2<br />
∫<br />
0 = P(N E ) dλ(E) = λ(N ω ) d P(ω) .<br />
E 1<br />
Thus, for almost all ω we have λ(N ω ) = 0. Consequently, by Corollary 9.4 there<br />
is no absolutely cont<strong>in</strong>uous spectrum <strong>in</strong> (E 1 , E 2 ) for these ω.<br />
□<br />
One might be tempted <strong>to</strong> th<strong>in</strong>k the assumption that all E ∈ [E 1 , E 2 ] are γ−good<br />
P-almost surely would imply that there are no generalized eigenvalues <strong>in</strong> [E 1 , E 2 ].<br />
This would exclude any spectrum <strong>in</strong>side (E 1 , E 2 ), not only absolutely cont<strong>in</strong>uous<br />
one. This reason<strong>in</strong>g is wrong. The problem with the argument is the follow<strong>in</strong>g:<br />
Under this assumption, we know that for any given energy E, there are no generalized<br />
eigenvalues with probability one, i.e. the set N E is a set of probability zero.<br />
Thus, for ω ∈ Ω 0 := ⋃ E∈[E 1 ,E 2 ] N E there are no generalized eigenvalues <strong>in</strong> the<br />
<strong>in</strong>terval [E 1 , E 2 ]. However, the set Ω 0 is an uncountable union of sets of measure<br />
zero, therefore, we cannot conclude that it has zero measure.<br />
Theorem 9.5 immediately triggers two k<strong>in</strong>d of questions: First, is (9.14) true under<br />
certa<strong>in</strong> assumptions, and how can we prove it This is exactly what the multiscale<br />
analysis does. We will discuss this result <strong>in</strong> the follow<strong>in</strong>g section 9.3 and prove it<br />
<strong>in</strong> chapter 10.<br />
The other question raised by the theorem is whether or not ‘good’ cubes might help<br />
<strong>to</strong> prove even pure po<strong>in</strong>t spectrum, not only the absence of absolutely cont<strong>in</strong>uous<br />
spectrum.<br />
It turns out that the condition (9.14) alone is not sufficient <strong>to</strong> prove pure po<strong>in</strong>t spectrum.<br />
There are examples of opera<strong>to</strong>rs with (almost periodic) potential V satisfy<strong>in</strong>g<br />
condition (9.14) <strong>in</strong>side their spectrum, hav<strong>in</strong>g no (l 2 -)eigenvalues at all (see e.g.<br />
[30]). So, these opera<strong>to</strong>rs have purely s<strong>in</strong>gular cont<strong>in</strong>uous spectrum <strong>in</strong> the region
where (9.14) holds. This effect is due <strong>to</strong> some k<strong>in</strong>d of ‘long range’ order of almost<br />
periodic potentials.<br />
In the situation of the <strong>An</strong>derson model, we have the <strong>in</strong>dependence of the random<br />
variables V ω (i). This assumption, we may hope, prevents the potential from ‘conspir<strong>in</strong>g’<br />
aga<strong>in</strong>st pure po<strong>in</strong>t spectrum through long range correlations. However, the<br />
above example of an almost periodic potential makes clear that some extra work is<br />
required <strong>to</strong> go beyond the absence of a.c. spectrum and prove pure po<strong>in</strong>t spectrum.<br />
This question will be addressed <strong>in</strong> section 9.5 after some preparation <strong>in</strong> section 9.4.<br />
9.3. The results of multiscale analysis .<br />
We def<strong>in</strong>e a length scale L k <strong>in</strong>ductively. The <strong>in</strong>itial length L 0 will be def<strong>in</strong>ed later<br />
depend<strong>in</strong>g on the specific parameters (disorder, energy region, etc.) of the problem<br />
considered. The length L k+1 is def<strong>in</strong>ed by L k α for an α with 1 < α < 2 <strong>to</strong> be<br />
further specified later. The constant α will only depend on some general parameters<br />
like the dimension d. The condition α > 1 ensures that L k → ∞, while α < 2<br />
makes the estimates <strong>to</strong> come easier. F<strong>in</strong>ally, we will have <strong>to</strong> choose α close <strong>to</strong> one.<br />
Observe, that the length scale L k is grow<strong>in</strong>g very fast, <strong>in</strong> fact superexponentially.<br />
A ma<strong>in</strong> result of multiscale analysis will be the follow<strong>in</strong>g probabilistic estimate,<br />
which holds for certa<strong>in</strong> <strong>in</strong>tervals [E 1 , E 2 ].<br />
RESULT 9.6 (multiscale analysis - weak form). For some α > 1, p > 2d and a<br />
γ > 0 and for all E ∈ I = [E 1 , E 2 ]<br />
85<br />
P ( Λ Lk is not (γ, E)−good for V ω ) ≤<br />
1 L p k<br />
. (9.19)<br />
REMARKS 9.7.<br />
(1) We will prove this result <strong>in</strong> the next two chapters.<br />
(2) To prove Result 9.6, we need <strong>to</strong> assume that the probability distribution<br />
P 0 of the random variables V ω (i) has a bounded density. This ensures<br />
that we can apply Wegner’s estimate (Theorem 5.23) which is a key <strong>to</strong>ol<br />
<strong>in</strong> our proof. Recently, Bourga<strong>in</strong> and Kenig [19] were able <strong>to</strong> do the<br />
multiscale analysis for some V ω without a density for P 0 .<br />
(3) We will proof Result 9.6 for I = [E 1 , E 2 ] when I is close <strong>to</strong> the bot<strong>to</strong>m<br />
of the spectrum or for given I if the disorder is sufficiently strong.<br />
(4) As the proof shows we have <strong>to</strong> take α < 2p<br />
2d+p<br />
which is bigger than 1<br />
s<strong>in</strong>ce p > 2d.<br />
The proof of Result 9.6 and its variants (see below) will take two chapters. We<br />
prove the result by <strong>in</strong>duction, i. e. we prove (9.19) for the <strong>in</strong>itial scale L 0 and then<br />
prove the <strong>in</strong>duction step, namely: If (9.19) holds for a certa<strong>in</strong> k, it holds for k + 1<br />
as well.<br />
The <strong>in</strong>itial scale estimate will be done <strong>in</strong> chapter 11. It is only here where we<br />
need assumptions about the energy <strong>in</strong>terval I (e.g. I is close <strong>to</strong> the bot<strong>to</strong>m of the<br />
spectrum or <strong>to</strong> an other band edge) or about the strength of the disorder. Thus, the<br />
specific parameters of the model enter only here.
86<br />
In contrast <strong>to</strong> this, the <strong>in</strong>duction step can be done under quite general conditions for<br />
all energies and any degree of disorder. This step will be presented <strong>in</strong> chapter 10.<br />
The multiscale estimate Result 9.6 obviously implies the absence of absolutely<br />
cont<strong>in</strong>uous spectrum <strong>in</strong>side I via Theorem 9.5. The estimate (9.19) per se does not<br />
imply pure po<strong>in</strong>t spectrum (see the discussion at the end of the previous section).<br />
However, for the <strong>An</strong>derson model one can use Result 9.6 <strong>to</strong> deduce pure po<strong>in</strong>t<br />
spectrum, provided P 0 has a bounded density. This can be done us<strong>in</strong>g a technique<br />
known as spectral averag<strong>in</strong>g. The basic idea goes back <strong>to</strong> Kotani [89] and was<br />
further developed and applied <strong>to</strong> the <strong>An</strong>derson model by various authors (see e.g.<br />
[37, 90, 126, 27]). The paper [126] triggered also the development of the theory of<br />
rank one perturbations [124]. We will not discuss this method here and refer <strong>to</strong> the<br />
papers cited.<br />
Instead, we will present another proof of pure po<strong>in</strong>t spectrum which goes back <strong>to</strong><br />
[46] and [40]. It consists <strong>in</strong> a version of the estimate (9.19) which is ‘uniform’ <strong>in</strong><br />
energy E. Taken literally a uniform version of (9.19) would be<br />
P ( There is an E ∈ I such that Λ Lk is not (γ, E)−good for V ω ) ≤ L −p<br />
k<br />
. (9.20)<br />
However, it is easy <strong>to</strong> see by <strong>in</strong>spect<strong>in</strong>g the proof of Theorem 9.5 that (9.20) implies<br />
that any E ∈ I is γ-good, thus there is no spectrum <strong>in</strong>side I by Corollary 9.3 above.<br />
In other words: condition (9.20) is ‘<strong>to</strong>o strong’ <strong>to</strong> imply pure po<strong>in</strong>t spectrum.<br />
A way out of this dilemma is <strong>in</strong>dicated by the ‘uniform’ version of Wegner’s estimate<br />
(Theorem 5.27). There, uniformity <strong>in</strong> energy is required only for pairs of<br />
disjo<strong>in</strong>t cubes. This leads us <strong>to</strong> a uniform version of Result 9.6 for pairs of cubes.<br />
RESULT 9.8 (multiscale analysis - strong form). For some p > 2d, an α with<br />
1 < α < 2p<br />
p+2d and a γ > 0 we have: For any disjo<strong>in</strong>t cubes Λ 1 = Λ Lk (n) and<br />
Λ 2 = Λ Lk (m)<br />
P ( For some E ∈ I both Λ 1 and Λ 2 are not (γ, E)−good ) ≤ L −2p<br />
k<br />
. (9.21)<br />
The proof of this result is an <strong>in</strong>duction procedure analogous <strong>to</strong> the one discussed<br />
above. In fact, the <strong>in</strong>itial step will be the same as for Result 9.6, see Chapter 11.<br />
In the <strong>in</strong>duction step we assume the validity of estimate (9.21) for k and deduce<br />
the assertion for k + 1 from this assumption. The general idea of this step is quite<br />
close <strong>to</strong> the <strong>in</strong>duction step for the weaker version (9.19), but it is technically more<br />
<strong>in</strong>volved. Therefore, we present the proof of the weak version first and then discuss<br />
the necessary changes for the strong (‘uniform’) version.<br />
9.4. <strong>An</strong> iteration procedure.<br />
One of the crucial <strong>in</strong>gredients of multiscale analysis is the observation that the<br />
estimate (9.11) <strong>in</strong> the proof of Theorem 9.2 can be iterated.<br />
A first version of this procedure is the contents of the follow<strong>in</strong>g result.<br />
We say that a subset A ⊂ Z d is well <strong>in</strong>side a set Λ (A ⋐ Λ) if A ⊂ Λ and<br />
A ∩ ∂ − Λ = ∅. For any set Λ ⊂ Z d we def<strong>in</strong>e the collection of L−cubes <strong>in</strong>side Λ
87<br />
by<br />
C L (Λ) = {Λ L (n) | Λ L (n) ⋐ Λ} . (9.22)<br />
We also set ∂ − L Λ = {m ∈ Λ | dist(m, ∂− Λ) ≤ L} (where dist(m, A) =<br />
<strong>in</strong>f k∈A || m − k|| ∞ ).<br />
THEOREM 9.9. Suppose that each cube <strong>in</strong> C M (A), A ⊂ Z d f<strong>in</strong>ite, is (γ, E)-good<br />
and M is large enough. If ψ is a solution of Hψ = Eψ <strong>in</strong> A and n 0 ∈ A with<br />
then<br />
for some γ ′ > 0.<br />
dist(n 0 , ∂ − A) ≥ k(M + 1) , (9.23)<br />
| ψ(n 0 )| ≤ e −γ′ kM sup<br />
m ∈∂ − M Λ | ψ(m)| (9.24)<br />
REMARK 9.10. Let us set<br />
and<br />
such that<br />
r = 2d (2M + 1) d−1 e −γM (9.25)<br />
γ ′ = γ − 1 M ln (2d (2M + 1) d−1) (9.26)<br />
r = e −γ′M . (9.27)<br />
Then the phrase ‘M large enough’ <strong>in</strong> the theorem means that r < 1 and the theorem<br />
holds with γ ′ as <strong>in</strong> (9.26). Note that γ ′ < γ, but the ‘error’ term γ − γ ′ = 1 M<br />
(<br />
(d −<br />
1) ln(2M + 1) + ln 2d) decreases <strong>in</strong> M and goes <strong>to</strong> zero if M tends <strong>to</strong> <strong>in</strong>f<strong>in</strong>ity.<br />
The theorem may look a bit clumsy at first sight. Nevertheless, it conta<strong>in</strong>s some of<br />
the ma<strong>in</strong> ideas of multiscale analysis. The estimate (9.24) says that any solution ψ<br />
decays exponentially <strong>in</strong> regions which are filled with good cubes. In other words:<br />
The tunnel<strong>in</strong>g probability of a quantum particle through such a region is exponentially<br />
small. This will f<strong>in</strong>ally lead <strong>to</strong> the <strong>in</strong>duction step <strong>in</strong> multiscale analysis.<br />
To illustrate Theorem 9.9, we state the follow<strong>in</strong>g Corollary which is essentially a<br />
reformulation of the theorem. The Corollary follows immediately from the Theorem.<br />
COROLLARY 9.11. Suppose each cube <strong>in</strong> C M (A), A ⊂ Z d f<strong>in</strong>ite, is (γ, E)-good<br />
and M ≥ C is large enough. Take n 0 ∈ A with d(n 0 ) = dist(n 0 , ∂ − A) so large<br />
≥ D. If ψ is a solution of Hψ = Eψ <strong>in</strong> A, then<br />
that d(n 0)<br />
M<br />
| ψ(n 0 )| ≤ e −γ′′ d(n 0 )<br />
sup | ψ(m)| (9.28)<br />
m ∈∂ − M Λ
88<br />
with<br />
γ ′′ = γ − 1 M ln ( 2d (2M + 1) d−1) ( 1 − 1 C − 1 D<br />
)<br />
. (9.29)<br />
Observe, that the error term 1 M ln ( 2d (2M + 1) d−1) ( 1 − 1 C − 1 D<br />
)<br />
is small if both<br />
M and the ratio of d(n 0 ) and M are big.<br />
PROOF (Theorem) : S<strong>in</strong>ce n 0 ∈ A and dist(n 0 , ∂ − A) ≥ (M + 1) we have<br />
Λ M (n 0 ) ∈ C M (A).<br />
Thus by (9.10), we have<br />
|ψ(n 0 )|<br />
≤<br />
∑<br />
(q,q ′ )∈∂Λ M (n 0 )<br />
q∈ Λ M (n 0 )<br />
|G Λ M (n 0 )<br />
E<br />
(n 0 , q)||ψ(q ′ )|<br />
≤ | ∂Λ M (n 0 )| e −γM sup |ψ(q ′ )|<br />
q ′ ∈∂ + Λ M (n 0 )<br />
≤ 2d (2M + 1) d−1 e −γM |ψ(n 1 )|<br />
= r |ψ(n 1 )| (9.30)<br />
for some n 1 ∈ ∂ + Λ M (n 0 ).<br />
If n 1 ∈ ∂ − M A, this is estimate (9.24) for k = 1. Note that dist(n 1, ∂ − M A) ≥<br />
dist(n 0 , ∂ − MA) − (M + 1).<br />
So n 1 ∈ ∂ − MA can only happen if k = 1.<br />
If n 1 ∉ ∂ − M A, we have Λ M(n 1 ) ∈ C M (A) and we can iterate the estimate (9.30)<br />
<strong>to</strong> obta<strong>in</strong><br />
with some n 2 ∈ ∂ + Λ M (n 1 ), so<br />
|ψ(n 1 )| ≤ r|ψ(n 2 )| (9.31)<br />
|ψ(n 0 )| ≤ r 2 |ψ(n 2 )| . (9.32)<br />
(Note, that for n 1 ∈ ∂ − MA the iteration might get us out of A!)<br />
For n 2 we have<br />
dist(n 2 , ∂ − M A) ≥ dist(n 1, ∂ − MA) − (M + 1)<br />
≥ dist(n 0 , ∂ − MA) − 2(M + 1) .<br />
So n 2 ∈ ∂ − MΛ can happen only if k ≤ 2.<br />
If n 2 ∉ ∂ − MΛ, then we may iterate (9.30) aga<strong>in</strong>. We obta<strong>in</strong><br />
|ψ(n 0 )| ≤ r|ψ(n 1 )| ≤ r 2 |ψ(n 2 )| ≤ r 3 |ψ(n 3 )| ≤ . . . ≤ r l |ψ(n l )| .<br />
This iteration process works f<strong>in</strong>e as long as the new po<strong>in</strong>t n l ∉ ∂ − M<br />
A. Consequently,<br />
by the assumption on n 0 , we can iterate at least k times.<br />
Thus, we obta<strong>in</strong>
89<br />
with some k ′ ≥ k.<br />
We conclude<br />
|ψ(n 0 )| ≤ r k′<br />
sup |ψ(q)| (9.33)<br />
q∈∂ − M Λ<br />
−γ′ kM<br />
|ψ(n 0 )| ≤ e sup |ψ(q)| . (9.34)<br />
q∈∂ − M Λ<br />
□<br />
REMARK 9.12. For ψ(n 0 ) ≠ 0, the above iteration procedure must f<strong>in</strong>ally reach<br />
∂ − L<br />
. Otherwise, we have<br />
|ψ(n<br />
0 )| ≤ r l sup |ψ(q)|<br />
q∈Λ<br />
for any l ∈ N which implies ψ(n 0 ) = 0. For ψ(n 0 ) = 0 the theorem is trivially<br />
fulfilled.<br />
9.5. From multiscale analysis <strong>to</strong> pure po<strong>in</strong>t spectrum.<br />
In this section we prove that the strong version (Result 9.8) of the multiscale estimate<br />
implies pure po<strong>in</strong>t spectrum <strong>in</strong>side the <strong>in</strong>terval where the estimate holds.<br />
THEOREM 9.13. If Result 9.8 holds for an <strong>in</strong>terval I = [E 1 , E 2 ], then with probability<br />
one<br />
σ c (H ω ) ∩ (E 1 , E 2 ) = ∅ .<br />
The spectrum of H ω <strong>in</strong>side (E 1 , E 2 ) consists of pure po<strong>in</strong>t spectrum, the correspond<strong>in</strong>g<br />
eigenfunctions decay exponentially at <strong>in</strong>f<strong>in</strong>ity.<br />
REMARK 9.14. The theorem <strong>in</strong>cludes the case (E 1 , E 2 ) ∩ σ(H ω ) = ∅ but we will<br />
choose E 1 , E 2 such that there is some spectrum <strong>in</strong>side (E 1 , E 2 ) when we apply the<br />
theorem.<br />
PROOF:<br />
Step 1<br />
We beg<strong>in</strong> with a little geometry. As before we choose a sequence L k by sett<strong>in</strong>g<br />
L k = L α k−1 with an α > 1 and L 0 <strong>to</strong> be determ<strong>in</strong>ed later. We consider the cubes<br />
Λ Lk = Λ Lk (0) and annuli A k which cover the region between the boundaries of<br />
Λ Lk and Λ Lk+1 , more precisely<br />
A k = Λ 6Lk+1 \ Λ 3Lk . (9.35)<br />
So, n ∈ A k if || n|| ∞ ≤ 6L k+1 and || n|| ∞ > 3L k . It is clear that<br />
A k ∩ A k+1 ≠ ∅ (9.36)
90<br />
and<br />
⋃<br />
Ak = Z d \ Λ 3L0 . (9.37)<br />
We will need also an enlarged version A + k of the A k namely<br />
A + k = Λ 8L k+1<br />
\ Λ 2Lk . (9.38)<br />
Obviously, A k ⊂ A + k and any n ∈ A k has a certa<strong>in</strong> ‘security’ distance from ∂ A + k ,<br />
<strong>in</strong> fact we have:<br />
LEMMA 9.15. For each n ∈ A k<br />
dist(n, ∂A + k ) ≥ 1 3 || n|| ∞ .<br />
PROOF (Lemma) :<br />
If || n|| ∞ ≥ 3L k we have<br />
dist(n, ∂Λ 2Lk ) = || n|| ∞ − 2L k<br />
≥<br />
|| n|| ∞ − 2 3 || n|| ∞<br />
= 1 3 || n|| ∞ .<br />
If || n|| ∞ ≤ 6L k+1<br />
dist(n, ∂Λ 8Lk+1 ) = 8L k+1 − || n|| ∞<br />
≥<br />
8 6 || n|| ∞ − || n|| ∞<br />
= 1 3 || n|| ∞ .<br />
If n ∈ A k we have 3L k ≤ || n|| ∞ ≤ 6L k+1 , so<br />
dist(n, ∂A + k ) = m<strong>in</strong>{dist(n, ∂Λ 6L k+1<br />
), dist(n, ∂Λ 3Lk )}<br />
≥ 1 3 || n|| ∞ .<br />
□<br />
Step 2<br />
Now, we <strong>in</strong>vestigate the probability that Λ Lk is not (E, γ)-good and, at the same<br />
time, one of the L k -cubes <strong>in</strong> A k is also not (E, γ)-good.<br />
Let us abbreviate<br />
C + k = C L k<br />
(A + k ) = {Λ L k<br />
(m)|Λ Lk (m) ⋐ A + k } .<br />
For a given k, def<strong>in</strong>e p k <strong>to</strong> be the probability of the event<br />
B k = { ω ∣ ∣ For some E ∈ [E 1 , E 2 ] , Λ Lk and at least one cube <strong>in</strong> C + k are not (E, γ)-good } .
91<br />
We will prove<br />
LEMMA 9.16. If Result 9.8 holds for I = [E 1 , E 2 ], then there is a constant C such<br />
that for all k<br />
p k<br />
≤<br />
C<br />
L 2p−αd<br />
k<br />
REMARK 9.17. The constants α and p are given <strong>in</strong> Result 9.8.<br />
. (9.39)<br />
PROOF (Lemma) : If Λ Lk (m) is a fixed cube <strong>in</strong> C + k then<br />
P ( For some E ∈ [E 1 , E 2 ] both Λ Lk (m) and Λ Lk are not (E, γ)-good )<br />
Hence<br />
S<strong>in</strong>ce α < 2p d<br />
≤ 1<br />
L 2p . (9.40)<br />
k<br />
P (B k ) ≤ | C + k | 1<br />
L 2p<br />
k<br />
≤ C (L α 1<br />
k )d L 2p<br />
k<br />
≤ C L d k+1<br />
≤<br />
C<br />
L 2p−αd<br />
k<br />
1<br />
L 2p<br />
k<br />
. (9.41)<br />
(by Result 9.8) we have 2p − αd > 0. Thus<br />
∑<br />
P (B k ) < ∞ . (9.42)<br />
Hence, by the Borel-Cantelli-Lemma (Theorem 3.6), we have<br />
Thus we have shown<br />
k<br />
P ( {ω | ω ∈ B k for <strong>in</strong>f<strong>in</strong>itely many k } ) = 0 . (9.43)<br />
PROPOSITION 9.18. If Result 9.8 holds for I = [E 1 , E 2 ], then for P-almost all ω,<br />
there is a k 0 = k 0 (ω) such that for all k ≥ k 0 :<br />
For any E ∈ [E 1 , E 2 ] either Λ Lk is (E, γ)-good or all cubes Λ Lk (m) <strong>in</strong> C + L k<br />
are<br />
(E, γ)-good.<br />
Step 3<br />
In this f<strong>in</strong>al step, we take ω such that the assertion of Proposition 9.18 is true.<br />
Suppose now that E ∈ [E 1 , E 2 ] is a generalized eigenvalue. It follows from Theorem<br />
9.2 that there is no sequence L ′ k (with L′ k → ∞) such that all Λ L ′ are (E, γ)-<br />
k<br />
good. Hence by Proposition 9.18, we conclude that for all k > k 1 , all cubes <strong>in</strong> C + k<br />
are (E, γ)-good.<br />
Let ψ be a generalized eigenfunction correspond<strong>in</strong>g <strong>to</strong> the generalized eigenvalue<br />
E. Take any n ∈ Z d with || n|| ∞ large enough. Then there is a k, k ≥ k 1 , so<br />
□
92<br />
that n ∈ A k (hence 3L k ≤ || n|| ∞ < 6L k+1 ). It follows from Lemma 9.15 that<br />
dist(n, ∂A + k ) ≥ 1 3 || n|| ∞. Thus we may apply theorem 9.9 <strong>to</strong> conclude<br />
| ψ(n)| ≤ e −γ′′ || n|| ∞<br />
sup<br />
m∈A + k<br />
| ψ(m)| . (9.44)<br />
S<strong>in</strong>ce ψ is polynomially bounded by assumption, we have for m ∈ A + k<br />
some r<br />
Thus<br />
| ψ(m)| ≤ C 0 (8L k+1 ) r<br />
≤<br />
≤<br />
C 1 L αr<br />
k<br />
C 2 || n|| αr<br />
∞ .<br />
and for<br />
| ψ(n)| ≤ e −˜γ|| n||∞ . (9.45)<br />
We have therefore shown that any generalized eigenfunction of H ω with eigenvalues<br />
<strong>in</strong> [E 1 , E 2 ] decays exponentially fast. A fortiori, any generalized eigenfunction<br />
is l 2 , so the correspond<strong>in</strong>g generalized eigenvalue is a bona fide eigenvalue. Thus,<br />
the spectrum <strong>in</strong> (E 1 , E 2 ) is pure po<strong>in</strong>t.<br />
□<br />
REMARK 9.19. Observe that eigenfunctions ψ 1 , ψ 2 <strong>to</strong> different eigenvalues are<br />
orthogonal <strong>to</strong> each other. S<strong>in</strong>ce the Hilbert space l 2 (Z d ) is separable, there are<br />
only countably many E ∈ [E 1 , E 2 ] with exponentially decay<strong>in</strong>g eigensolutions.<br />
Notes and Remarks<br />
multiscale analysis is based on the ground break<strong>in</strong>g paper by Fröhlich and Spencer<br />
[47]. That the MSA result implies absence of a.c. spectrum was realized by Mart<strong>in</strong>elli<br />
and Scoppolla [101]. <strong>An</strong> alternative appoach <strong>to</strong> exclude a.c. spectrum can<br />
be found <strong>in</strong> [125].<br />
The first proofs of <strong>An</strong>derson localization were given <strong>in</strong>dependently <strong>in</strong> [46], [37],<br />
[126]. The latter papers develop the method of spectral averag<strong>in</strong>g which goes partly<br />
back <strong>to</strong> [89].<br />
The method <strong>to</strong> prove <strong>An</strong>derson localization we present above is due <strong>to</strong> [40] which<br />
is related <strong>to</strong> [46]. Germ<strong>in</strong>et and Kle<strong>in</strong> [50] <strong>in</strong>vestigate the relation between Localization<br />
and multiscale analysis <strong>in</strong> great detail. They characterize a certa<strong>in</strong> version<br />
of localization <strong>in</strong> terms of the multiscale estimate.<br />
For the literature on the cont<strong>in</strong>uous case, i.e. for Schröd<strong>in</strong>ger opera<strong>to</strong>rs on L 2 (R d ),<br />
we refer <strong>to</strong> the Notes at the end of the next chapter.
93<br />
10. Multiscale analysis<br />
10.1. Strategy.<br />
We turn <strong>to</strong> the proof of the multiscale analysis result.<br />
Multiscale analysis (MSA) is an <strong>in</strong>duction procedure which starts with a certa<strong>in</strong><br />
length scale L 0 and then proves the validity of the multiscale estimate (9.6 and 9.8)<br />
for L k+1 = L α k assum<strong>in</strong>g the estimate holds for L k. The value of α will be fixed<br />
later. To get an <strong>in</strong>creas<strong>in</strong>g sequence L k we obviously need α > 1. We will also<br />
choose α < 2 for reasons that will become clear later. In fact, later we will have <strong>to</strong><br />
choose α close <strong>to</strong> one.<br />
In this chapter we will present the <strong>in</strong>duction step (from L k <strong>to</strong> L k+1 ) deferr<strong>in</strong>g the<br />
<strong>in</strong>itial step (for L 0 ) <strong>to</strong> the next chapter. The <strong>in</strong>duction step can be done for all<br />
energies E and for arbitrary degree of disorder (provided there is some disorder,<br />
of course). Thus, it is the <strong>in</strong>itial step which dist<strong>in</strong>guishes between energy regions<br />
with pure po<strong>in</strong>t spectrum and those energies where we might have (absolutely)<br />
cont<strong>in</strong>uous spectrum. As expla<strong>in</strong>ed <strong>in</strong> chapter 8, we expect certa<strong>in</strong> energy regions<br />
with absolutely cont<strong>in</strong>uous spectrum, but are not (yet) able <strong>to</strong> prove it.<br />
The proof of the <strong>in</strong>duction step consists of an analytical and a probabilistic part.<br />
We start with analytic estimates.<br />
For the rest of this chapter, we set for brevity l = L k and L = L k+1 , so we do<br />
the <strong>in</strong>duction step from l <strong>to</strong> L = l α . By tak<strong>in</strong>g L 0 sufficiently large we can always<br />
assume that l and, a fortiori, L is big enough, i. e. bigger than a certa<strong>in</strong> constant.<br />
S<strong>in</strong>ce α > 1 we have L ≫ l. Below, we will need that both l and L are <strong>in</strong>tegers. To<br />
ensure this we should actually choose L <strong>to</strong> be the smallest <strong>in</strong>teger bigger or equal<br />
<strong>to</strong> l α . We will neglect this po<strong>in</strong>t, it would complicate the notation. However, the<br />
reason<strong>in</strong>g of the proof rema<strong>in</strong>s the same.<br />
The analytic estimate is a puzzle with different types of cubes. There are (small)<br />
cubes Λ l (r) of size l = L k and (big) cubes Λ L (m) of size L = L k+1 = l α .<br />
The goal is <strong>to</strong> prove that that the Green’s function (H ΛL − E) −1 (m, n) decays<br />
exponentially.<br />
By <strong>in</strong>duction hypothesis the probability that a small cube (of size l) is (γ, E)-good<br />
is very high. Thus, we expect that most of the small cubes Λ l (n) <strong>in</strong>side Λ L are<br />
(γ, E)−good. Let us suppose for the moment, that actually all cubes of size l<br />
<strong>in</strong>side Λ L are (γ, E)-good. Then, us<strong>in</strong>g the geometric resolvent identity (5.53) and<br />
iterat<strong>in</strong>g it just as we did <strong>in</strong> the proof of Theorem 9.9 will give us an estimate for<br />
the Green’s function G Λ L<br />
E<br />
of the form<br />
|G Λ L<br />
E (n, m)| ≤ e−˜γkl |G Λ L<br />
E (n k, m)|. (10.1)<br />
This estimate results from apply<strong>in</strong>g the geometric resolvent equation k times. This<br />
step can be iterated as long as the po<strong>in</strong>t n k is not <strong>to</strong>o close <strong>to</strong> the boundary of<br />
Λ L (so that the cube of size l around n k belongs <strong>to</strong> Λ L ) and the cube Λ l (n k ) is a<br />
(γ, E)-good cube. If all cubes of size l <strong>in</strong>side Λ L are good, we expect that we can<br />
iterate roughly L l<br />
times before we reach the boundary and conclude
94<br />
|G Λ L<br />
E (n, m)| ≤ e−˜γL |G Λ L<br />
E (n′ , m)|. (10.2)<br />
We may hope that we can obta<strong>in</strong> an estimate of the type (10.2) even if not all<br />
l-cubes <strong>in</strong> Λ L are good but, at least, an overwhelm<strong>in</strong>g majority of them is.<br />
Once we have (10.2) we need a rough a priori bound on G Λ L<br />
E (n′ , m) <strong>to</strong> obta<strong>in</strong> the<br />
desired exponential estimate for G Λ L<br />
E (n, m), i. e. we need <strong>to</strong> know that Λ L is not<br />
an extremely bad cube. We say that a cube is extremely bad, if it is resonant <strong>in</strong> the<br />
sense of the follow<strong>in</strong>g def<strong>in</strong>ition.<br />
DEFINITION 10.1. We call a cube Λ L (n) E-resonant if dist(E, σ(H ΛL (n))) <<br />
e −√L .<br />
From Wegner’s estimate (Theorem 5.23) we immediately learn that it is very unlikely<br />
(at least for large L) that a cube is E-resonant, <strong>in</strong> fact<br />
PROPOSITION 10.2. If the (s<strong>in</strong>gle-site) measure P 0 has a bounded density, then<br />
P(Λ L (n) is E-resonant) ≤ C (2L + 1) d e −√L . (10.3)<br />
If Λ L (n) is not E-resonant, we know that the Green’s function G Λ L(n)<br />
E<br />
exists, because<br />
E is not <strong>in</strong> the spectrum. We even have a rough estimate on the Green’s<br />
function which tells us that Λ L is not ‘extremely bad’.<br />
PROPOSITION 10.3. If the cube Λ L (n) is not E-resonant, then for all<br />
m, m ′ ∈ Λ L (n)<br />
|G Λ L(n)<br />
E<br />
(m, m ′ )| ≤ e√<br />
L<br />
. (10.4)<br />
PROOF:<br />
If Λ L is not E-resonant then<br />
|G Λ L<br />
E (m, m′ )| = |(H ΛL − E) −1 (m, m ′ )|<br />
≤ || (H ΛL − E) −1 ||<br />
≤<br />
1<br />
dist(E, σ(H ΛL ))<br />
≤ e√<br />
L<br />
. (10.5)<br />
□<br />
Thus, if the cube Λ L is not resonant and if we have (10.2), we get an estimate of<br />
the form<br />
|G Λ L<br />
E (n, m)| ≤ e−˜γL e√<br />
L<br />
(10.6)<br />
≤ e −γ′L . (10.7)<br />
What we f<strong>in</strong>ally shall prove <strong>in</strong> (the analytical part of) the <strong>in</strong>duction step is:
95<br />
If an overwhelm<strong>in</strong>g majority of the cubes Λ l (m) <strong>in</strong> Λ L is (γ, E)−good and Λ L<br />
itself is not E-resonant, then Λ L is (γ ′ , E)−good.<br />
Note that the exponential rates differ. In fact, γ ′ < γ. That is <strong>to</strong> say, we can not<br />
avoid <strong>to</strong> decrease the decay rate <strong>in</strong> each and every <strong>in</strong>duction step. As a result we<br />
get a sequence of rates γ 0 , γ 1 , . . . (for <strong>in</strong>duction step 0, 1, . . . ) . Of course, if<br />
γ n → 0 (or becomes negative) the whole result is pretty useless. So, we have <strong>to</strong><br />
prove that γ n ↘ γ ∞ > 0.<br />
Once we have an analytic estimate of the above type, the <strong>in</strong>duction step will be<br />
completed by a probabilistic estimate. We have <strong>to</strong> prove that with high probability<br />
most cubes Λ l (j) <strong>in</strong>side of Λ L are (γ, E)-good and Λ L is not E-resonant. This<br />
probability has <strong>to</strong> be bigger than 1−L −p . To prove that most cubes Λ l (j) are good,<br />
we use the <strong>in</strong>duction hypothesis. That Λ L is not resonant with high probability<br />
follows from the Wegner estimate Theorem 5.23.<br />
We have deliberately used the vague terms ‘most cubes’ and ‘an overwhelm<strong>in</strong>g<br />
majority’. What they exactly mean is yet <strong>to</strong> be def<strong>in</strong>ed.<br />
10.2. <strong>An</strong>alytic estimate - first try .<br />
We start with a first attempt <strong>to</strong> do the analytic part of the <strong>in</strong>duction step. This first<br />
try assumes that all cubes of size l <strong>in</strong>side Λ L are (γ, E)−good. We recall that<br />
C l (Λ L ) = {Λ l (m) | Λ l (m) ⋐ Λ L }.<br />
The ma<strong>in</strong> idea of the approach is already conta<strong>in</strong>ed <strong>in</strong> the proof of Theorem 9.9.<br />
PROPOSITION 10.4. Suppose all cubes <strong>in</strong> C l (Λ L ) are (γ, E)−good. Then for any<br />
¯γ < γ there is an l 0 such that for l ≥ l 0<br />
|G Λ L<br />
E (m, n)| = |(H Λ L<br />
− E) −1 (m, n)| ≤<br />
for any m ∈ Λ L 1/2 and any n ∈ ∂ − Λ L .<br />
1<br />
dist(E, σ(H ΛL )) e−¯γL (10.8)<br />
PROOF: Take m ∈ Λ L 1/2. S<strong>in</strong>ce dist(m, ∂ − Λ L ) ≥ l + 1 if l 0 and hence l is<br />
large enough, we have Λ l (m) ∈ C l (Λ L ) and we may apply the geometric resolvent<br />
equation (5.53). Thus, we have<br />
with<br />
|G Λ L<br />
E (m, n)| ≤ ∑<br />
for some n 1 ∈ ∂ + Λ l (m).<br />
(q,q ′ )∈∂Λ l (m)<br />
q∈ Λ l (m)<br />
|G Λ l(m)<br />
E<br />
(m, q)| |G Λ L<br />
E (q′ , n)| (10.9)<br />
≤ 2d (2l + 1) d−1 e −γl |G Λ L<br />
E (n 1, n)| (10.10)<br />
≤ e −˜γl |G Λ L<br />
E (n 1, n)| (10.11)<br />
˜γ = γ −<br />
(d − 1) ln(2l + 1)<br />
l<br />
−<br />
ln 2d<br />
l<br />
(10.12)
96<br />
If dist(n 1 , ∂ − Λ L ) ≥ l + 1, we may repeat this estimate with Λ l (m) replaced by<br />
Λ l (n 1 ) and obta<strong>in</strong><br />
|G Λ L<br />
E (m, n)| ≤ e−˜γ 2l |G Λ L<br />
E (n 2, n)|<br />
with n 2 ∈ ∂ + Λ l (n 1 ).<br />
Note that dist(n 1 , ∂ − Λ L ) ≥ L − √ L − (l + 1), s<strong>in</strong>ce n 1 ∈ ∂ + Λ l (m).<br />
So, the second estimation step is certa<strong>in</strong>ly possible if L − √ L − (l + 1) ≥ l + 1.<br />
If this is so, we may try <strong>to</strong> iterate (10.9) a second time. This is possible if<br />
and the result is<br />
L − √ L − 2(l + 1) ≥ l + 1<br />
|G Λ L<br />
E (m, n)| ≤ e−˜γ 3l |G Λ L<br />
E (n 3, n)| .<br />
We may apply this procedure k times as long as L − √ L − k(l + 1) ≥ l + 1, i. e.<br />
for<br />
k ≤<br />
The largest <strong>in</strong>teger k 0 satisfy<strong>in</strong>g (10.13) is at least<br />
L<br />
√<br />
L<br />
l + 1 − l + 1 − 1 . (10.13)<br />
Consequently, we obta<strong>in</strong><br />
k 0 ≥<br />
L<br />
l + 1 − √<br />
L<br />
l + 1 − 2 . (10.14)<br />
As long as ˜γ > 0, we have<br />
|G Λ L<br />
E (m, n)| ≤ e−˜γ k 0l |G Λ L<br />
E (n k 0<br />
, n)|<br />
≤<br />
=<br />
||(H ΛL − E) −1 || e −˜γ k 0l<br />
1<br />
e−˜γ k0l . (10.15)<br />
dist(E, σ(H Λ ))<br />
e −˜γk 0l<br />
≤<br />
e −˜γ(L l<br />
l+1 −√ L l<br />
l+1 −2l)<br />
= e −˜γ( l<br />
l+1 − √ 1<br />
L<br />
≤ e −˜γ(1− 1<br />
l+1 − √ 1<br />
L<br />
l<br />
l+1 −2 l L )L<br />
l<br />
l+1 −2 l L )L<br />
≤ e −˜γ(1− 1 l − 1 1<br />
−2<br />
lα/2 l α−1 )L . (10.16)<br />
So, estimate (10.8) holds if<br />
(<br />
(d − 1) ln(2l + 1)<br />
γ − − 2d l<br />
l<br />
By tak<strong>in</strong>g l large enough we can assure that (10.17) holds.<br />
) (<br />
1 − 1 l − 1<br />
l α/2 − 2 1<br />
l α−1 )<br />
≥ ¯γ . (10.17)<br />
If we assume that Λ L is not E-resonant (see Def<strong>in</strong>ition 10.1), we can further estimate<br />
expression (10.8).<br />
□
THEOREM 10.5. If the cube Λ L is not E-resonant and if all the cubes <strong>in</strong> C l (Λ L )<br />
are (γ, E)−good and γ ′ < γ, then<br />
if l is large enough.<br />
PROOF:<br />
obta<strong>in</strong><br />
with γ ′ = ¯γ − 1<br />
l α/2 .<br />
Λ L is (γ ′ , E)−good<br />
By (10.8) and the assumption that Λ L is not resonant (see 10.5) we<br />
|G Λ L<br />
E (m, n)| ≤ e−¯γ L e L1/2<br />
≤<br />
e −γ′ L<br />
97<br />
(10.18)<br />
□<br />
COROLLARY 10.6. If the cube Λ L is not E-resonant and if all the cubes <strong>in</strong> C l (Λ L )<br />
are (γ, E)−good, then Λ L is (γ ′ , E)−good with<br />
γ ′ ≥ γ ( 1 − 4 ) (3d ln(2l + 1)<br />
−<br />
l α−1 + 1 )<br />
. (10.19)<br />
l l α/2<br />
Moreover, for l ≥ C 0 , with C 0 depend<strong>in</strong>g only on α and the dimension d, we have<br />
γ ′ ≥ γ ( 1 − 4 ) 2 −<br />
l α−1 . (10.20)<br />
lα/2 PROOF:<br />
Estimate (10.19) follows from (10.17), (10.18) and the observation that<br />
α − 1 < α/2 < 1 (10.21)<br />
s<strong>in</strong>ce 1 < α < 2.<br />
Moreover, there is a constant C 0 = C 0 (α, d) such that for l ≥ C 0 we have<br />
3d ln(2l+1)<br />
l<br />
≤ 1 which implies (10.20).<br />
□<br />
l α/2<br />
<strong>An</strong> obvious problem with the above result is the fact that we have <strong>to</strong> decrease<br />
the rate γ of the exponential decay <strong>in</strong> each <strong>in</strong>duction step. Suppose we start with<br />
a rate γ 0 for length scale L 0 . Let us assume L 0 ≥ C 0 , the constant appear<strong>in</strong>g<br />
before (10.20). We call γ k ≤ γ 0 the decay rate we obta<strong>in</strong> from Theorem 10.5 and<br />
Corollary 10.6 <strong>in</strong> the k th step, i. e. for L k = (L k−1 ) α .<br />
We get the lower bound<br />
4<br />
γ k+1 ≥ γ k − γ k α−1<br />
L − 2<br />
(10.22)<br />
α/2<br />
k L k<br />
Thus<br />
γ ∞ = lim <strong>in</strong>f γ k ≥ γ 0 − γ 0 ∞ ∑<br />
≥ γ k − γ 0<br />
4<br />
L k<br />
α−1 − 2<br />
L k<br />
α/2 . (10.23)<br />
k=0<br />
4<br />
α−1<br />
L − ∑<br />
∞<br />
k<br />
k=0<br />
2<br />
L k<br />
α/2 . (10.24)
98<br />
To estimate the right hand side of (10.24), we use the follow<strong>in</strong>g lemma.<br />
LEMMA 10.7. For β > 0 and L 0 large enough we have<br />
∞∑ 1<br />
β<br />
L ≤ 2<br />
β k L . (10.25)<br />
0<br />
k=0<br />
REMARK 10.8. In the lemma L 0 large means: L 0 β(α−1) ≥ 2.<br />
PROOF:<br />
r k := 1<br />
L β k<br />
≤<br />
≤<br />
1<br />
α<br />
(L k 0 ) β ≤ 1<br />
(L β 0 )αk<br />
1<br />
(L β 0 )1+k(α−1) ≤ 1<br />
L β 0<br />
Above, we used α k ≥ 1 + k(α − 1).<br />
From these estimates we obta<strong>in</strong> for L β(α−1) 0 ≥ 2<br />
∞∑<br />
1 1<br />
r := r k ≤<br />
−β(α−1)<br />
1 − L 0<br />
k=0<br />
( 1<br />
L β 0<br />
L β(α−1)<br />
0<br />
≤ 2<br />
L β 0<br />
) k . (10.26)<br />
. (10.27)<br />
From this lemma we learn that the ‘f<strong>in</strong>al’ decay rate γ ∞ is positive if L 0 and γ 0 are<br />
not <strong>to</strong>o small, more precisely:<br />
PROPOSITION 10.9. If L 0 is big enough and<br />
then<br />
REMARK 10.10. L 0 big enough means<br />
γ 0 ≥ 16<br />
L 0<br />
α/2<br />
□<br />
(10.28)<br />
γ ∞ = <strong>in</strong>f γ k ≥ 1 2 γ 0 . (10.29)<br />
L 0 α−1 ≥ 32 and L 0<br />
(α−1) 2 ≥ 2 . (10.30)<br />
PROOF: S<strong>in</strong>ce α < 2, we know α 2 ≥ (α − 1). So, if L 0 (α−1)2 ≥ 2, by Lemma<br />
10.7 we have<br />
∞∑ 1<br />
α/2<br />
L ≤ 2<br />
(10.31)<br />
α/2<br />
k L 0<br />
and<br />
k=0<br />
∞∑<br />
k=0<br />
1<br />
L k<br />
α−1 ≤ 2<br />
L 0<br />
α−1 . (10.32)
Thus, (10.28) and (10.30) <strong>in</strong>serted <strong>in</strong> (10.24) give<br />
8<br />
γ ∞ ≥ γ 0 −<br />
α−1<br />
L γ 0 −<br />
0<br />
Let us pause <strong>to</strong> summarize what we have done so far.<br />
4<br />
L 0<br />
α/2<br />
99<br />
(10.33)<br />
≥ 3 4 γ 0 − 1 4 γ 0 = 1 2 γ 0 . (10.34)<br />
THEOREM 10.11. Def<strong>in</strong>e the length scale L k+1 = L k α with 1 < α < 2 and a<br />
suitable L 0 , which is not <strong>to</strong>o small.<br />
If for a certa<strong>in</strong> k<br />
(1) all the cubes <strong>in</strong> C Lk (Λ Lk+1 ) are (γ k , E)- good and<br />
(2) the cube Λ Lk+1 is not E-resonant<br />
then the cube Λ Lk+1 is (γ k+1 , E)- good with a rate γ k+1 satisfy<strong>in</strong>g<br />
γ k+1 ≥ γ k − γ k<br />
4<br />
L k<br />
α−1 − 2<br />
L k<br />
α/2 . (10.35)<br />
Moreover, we have some control on the sequence γ k .<br />
COROLLARY 10.12. If the <strong>in</strong>itial rate γ 0 satisfies γ 0 ≥ 16<br />
enough, then the γ k (as <strong>in</strong> (10.35)) satisfy γ k ≥ γ 0<br />
2<br />
for all k.<br />
L 0<br />
α/2<br />
□<br />
and L 0 is large<br />
Thus, we have done a first version of the analytic part of the MSA-proof. So far<br />
for the good news about Theorem 10.11.<br />
We are left with the probabilistic estimates, namely:<br />
Prove that if Λ Lk is good with high probability then the hypothesis’ (1) and (2) <strong>in</strong><br />
Theorem 10.11 above are true with high probability. More precisely, we would like<br />
<strong>to</strong> prove:<br />
If<br />
then<br />
with L = l α .<br />
P(Λ l is not (γ, E)−good) ≤ 1 l p<br />
P(Λ L is not (γ, E)−good) ≤ 1 L p (10.36)<br />
Here comes the bad news: There is no chance for such an estimate.<br />
In fact, Theorem 10.11 allows us <strong>to</strong> estimate<br />
P(Λ L is not (γ, E)−good)<br />
≤ P(Λ L is not E-resonant) + P(at least one cube <strong>in</strong> C l (Λ L ) is not (γ, E)−good) .<br />
(10.37)
100<br />
The first term <strong>in</strong> (10.37) can be estimated by the Wegner estimate (5.23). However<br />
the second term is certa<strong>in</strong>ly bigger than P(Λ L (0) is not (γ, E)−good). The only<br />
estimate we have for this is 1<br />
l<br />
. So the best we can possibly hope for is an estimate<br />
p<br />
like<br />
P(Λ L is not (γ, E)−good) ≤ 1 l p = 1 . (10.38)<br />
Lp/α This is much worse than estimate (10.36).<br />
What goes wrong here is that the probability that all small cubes are good is <strong>to</strong>o<br />
small. Consequently, we have <strong>to</strong> accept at least one or even a few cubes <strong>in</strong> C l (Λ L )<br />
which are not (γ, E)−good. Deal<strong>in</strong>g with bad cubes <strong>in</strong> C l (Λ L ) requires a ref<strong>in</strong>ed<br />
version of the above analytic reason<strong>in</strong>g.<br />
10.3. <strong>An</strong>alytic estimate - second try.<br />
Now, we try <strong>to</strong> do the <strong>in</strong>duction step allow<strong>in</strong>g a few bad cubes <strong>in</strong> C l (Λ L ). We<br />
start with just one bad cube. More precisely, we suppose now that C l (Λ L ) does not<br />
conta<strong>in</strong> two disjo<strong>in</strong>t cubes which are not (γ, E)−good.<br />
If two cubes overlap, events connected with these cubes are not <strong>in</strong>dependent, so<br />
probability estimates are hard <strong>in</strong> this case. That is why we <strong>in</strong>sist above on non<br />
overlapp<strong>in</strong>g sets.<br />
The above assumption implies that there is an m 0 ∈ Λ L such that all the cubes<br />
Λ l (m) ∈ C l (Λ L ) with || m − m 0 || ∞ > 2l are (γ, E)−good. Consequently, there<br />
are no bad cubes with centers outside Λ 2l (m 0 ). The cube Λ 2l (m 0 ) is the ‘dangerous’<br />
region which requires special care.<br />
As <strong>in</strong> the proof of Proposition 10.4, we use and iterate the geometric resolvent<br />
equation <strong>to</strong> estimate<br />
|G Λ L<br />
E (m, n)| ≤ e−˜γ lr |G Λ L<br />
E (n r, n)| (10.39)<br />
as long as possible. With a bad cube <strong>in</strong>side Λ L , this procedure can s<strong>to</strong>p not only<br />
when n r is near the boundary of Λ L but also if n r reaches the problematic region<br />
around m 0 where cubes Λ l (m) might be bad.<br />
Let us concentrate for a moment how we can handle sites n r <strong>in</strong>side the dangerous<br />
region Λ 2l (m 0 ). So, suppose that u := n r ∈ Λ 2l (m 0 ). Hence we cannot be sure<br />
the cube Λ l (u) is good. We can still try <strong>to</strong> apply the geometric resolvent equation<br />
and obta<strong>in</strong><br />
∑<br />
|G Λ L<br />
E<br />
(u, n)| ≤<br />
(q,q ′ )∈∂Λ l (u)<br />
q∈ Λ l (u)<br />
|G Λ l(u)<br />
E<br />
(u, q)| |G Λ L<br />
E (q′ , n)| . (10.40)<br />
If we assume noth<strong>in</strong>g about the cube Λ l (u), there is no chance <strong>to</strong> estimate G Λ l(u)<br />
E<br />
(u, q).<br />
In fact, this Green’s function may be arbitrarily large or even non exist<strong>in</strong>g. It seems<br />
reasonable <strong>to</strong> suppose that the ‘trouble mak<strong>in</strong>g’ region, the cube Λ 2l (m 0 ), is ‘not<br />
completely bad’ <strong>in</strong> the sense, that Λ 2l (m 0 ) is not E-resonant. This allows us <strong>to</strong><br />
estimate
101<br />
|G Λ L<br />
E (u, n)| ≤ ∑<br />
(q,q ′ )∈∂Λ 2l (m 0 )<br />
q∈ Λ 2l (m 0 )<br />
|G Λ 2l(m 0 )<br />
E<br />
(u, q)| |G Λ L<br />
E (q′ , n)|<br />
≤ 2d (4l + 1) d−1 e√<br />
2l<br />
|G Λ L<br />
E (u′ , n)| (10.41)<br />
for a u ′ ∈ Λ L \Λ 2l (m 0 ).<br />
Observe, that the cube Λ l (u ′ ) is (γ l , E)−good by <strong>in</strong>duction hypothesis s<strong>in</strong>ce<br />
u ′ ∉ Λ 2l (m 0 ). Therefore, the next iteration of the geometric resolvent estimate<br />
will give us an exponentially decreas<strong>in</strong>g term<br />
|G Λ √<br />
L<br />
E (u, n)| ≤ 2d (4l + 1)d−1 e 2l<br />
|G Λ L<br />
E (u′ , n)|<br />
≤ (2d) 2 (4l + 1) d−1 (2l + 1) d−1 e√<br />
2l<br />
e −γ l l |G Λ L<br />
E (n r+1, n)| . (10.42)<br />
In the double step (10.41) and (10.42), we pick up a fac<strong>to</strong>r<br />
ρ := (2d) 2 (4l + 1) d−1 (2l + 1) d−1 e√<br />
2l<br />
e −γ l l . (10.43)<br />
The second step (10.42) compensates the first one (10.41) if ρ ≤ 1. This is the case<br />
if<br />
√<br />
2 2 ln (2d) + 2 (d − 1) ln (4l + 1)<br />
γ l ≥ √ + (10.44)<br />
l l<br />
which is fulfilled for<br />
γ l ≥ 2 √<br />
l<br />
(10.45)<br />
if l is bigger than a constant depend<strong>in</strong>g only on the dimension.<br />
In the proof of Theorem 10.5, we could choose (see (10.22))<br />
γ k+1 ≥ γ k − γ k<br />
4<br />
L k<br />
α−1 − 2<br />
L k<br />
α/2 . (10.46)<br />
<strong>An</strong> <strong>in</strong>duction argument us<strong>in</strong>g (10.46) shows<br />
LEMMA 10.13. If L 0 ≥ M, a constant depend<strong>in</strong>g only on α and d, and if (10.46)<br />
holds, then γ 0 ≥ 2 implies<br />
L 1/2<br />
0<br />
γ k ≥ 2<br />
L 1/2 for all k . (10.47)<br />
k<br />
This Lemma ensures that we can iterate the <strong>in</strong>duction step <strong>in</strong> the multiscale analysis<br />
even if we hit the dangerous region Λ 2l (m 0 ). In fact, once we start with γ 0 ≥<br />
2<br />
L 1/2<br />
, we can be sure that all the the rates satisfy the condition γ k ≥ 2<br />
0 L 1/2<br />
.<br />
k<br />
PROOF:<br />
By tak<strong>in</strong>g L 0 large enough we can ensure that:<br />
4<br />
L α−1<br />
k<br />
≤ 1 2<br />
and<br />
4<br />
L α/2<br />
k<br />
≤ 1<br />
L 1/2<br />
k<br />
for all k . (10.48)
102<br />
So, if γ k ≥ 2<br />
L 1/2<br />
k<br />
, then<br />
γ k+1 ≥ γ k (1 − 4<br />
L α−1<br />
k<br />
≥ 1 2 γ k − 2<br />
≥ 1<br />
L 1/2<br />
k<br />
≥ 2<br />
L α/2<br />
k<br />
≥ 2<br />
L 1/2<br />
k+1<br />
Thus, the Lemma follows by <strong>in</strong>duction.<br />
L α/2<br />
k<br />
− 2<br />
L α/2<br />
k<br />
) − 2<br />
L α/2<br />
k<br />
□<br />
Know<strong>in</strong>g how <strong>to</strong> deal with the cubes <strong>in</strong>side Λ 2l (m 0 ), we now sketch our strategy.<br />
We use the geometric resolvent equation <strong>to</strong> estimate the resolvent on the big cube<br />
of size L <strong>in</strong> terms of the resolvent of small cubes of size l. As long as the first<br />
argument n r of the Green’s function (for Λ L ) belongs <strong>to</strong> a good cube, we use an<br />
exponential bound as <strong>in</strong> (10.11). If n r belongs <strong>to</strong> the ‘bad’ region which may<br />
conta<strong>in</strong> cubes that are not good, then we do the double step estimate (10.41) and<br />
(10.42). This procedure can be repeated until we get close <strong>to</strong> the boundary of Λ L .<br />
The number of times we do the exponential bound <strong>in</strong> this procedure is at least of<br />
the order L/l. In fact, analogously <strong>to</strong> (10.13) the number k 0 of ‘good’ steps is at<br />
least<br />
k 0 ≥<br />
L<br />
√<br />
L<br />
l + 1 − l + 1 − C 1 . (10.49)<br />
Consequently, the estimates of the previous section can be redone if we allow ‘one’<br />
bad cube with the follow<strong>in</strong>g changes<br />
• We need L 0 ≥ C 2 with a constant C 2 (possibly) bigger than the previous<br />
one.<br />
• We have <strong>to</strong> take γ 0 ≥ 2 L 0<br />
−1/2<br />
• The procedure requires that all cubes of size 2l <strong>in</strong>side Λ L are non resonant.<br />
While we need this only for the cube Λ 2l (m 0 ) around the ‘bad’<br />
cube, we do not know, where the bad cube is, so we require non resonance<br />
for all cubes of the appropriate size.<br />
Thus, we have shown the follow<strong>in</strong>g improvement of Theorem 10.11.<br />
THEOREM 10.14. Suppose L 0 is large enough and L k+1 = L k α with 1 < α < 2.<br />
If for a certa<strong>in</strong> k (l := L k and L := L k+1 )<br />
(1) there do not exist two disjo<strong>in</strong>t cubes <strong>in</strong> C l (Λ L ) which are not (γ k , E)-<br />
good with a rate γ k ≥ 2<br />
l 1/2 ,
(2) no cube Λ 2l (m) <strong>in</strong> Λ L is E-resonant and<br />
(3) the cube Λ L is not E-resonant,<br />
then the cube Λ L is (γ k+1 , E)- good with a rate γ k+1 satisfy<strong>in</strong>g γ k+1 ≥ 2<br />
L 1/2 .<br />
Moreover we can choose the rate γ k+1 such that<br />
103<br />
γ k+1 ≥ γ k − γ k<br />
C<br />
L k<br />
α−1 − C<br />
L k<br />
α/2 . (10.50)<br />
As above, we can estimate the decay rates as follows.<br />
COROLLARY 10.15. If the <strong>in</strong>itial rate γ 0 satisfies γ 0 ≥ C<br />
L 0<br />
1/2<br />
enough, then the γ k <strong>in</strong> Theorem 10.14 satisfy γ k ≥ γ 0<br />
2<br />
for all k.<br />
and L 0 is large<br />
This result allows us <strong>to</strong> prove the multiscale estimate <strong>in</strong> its weak form (9.6) as we<br />
will show <strong>in</strong> the next section 10.4 where we do the correspond<strong>in</strong>g probabilistic<br />
estimates.<br />
The above analytic results (especially the counterpart of Theorem 10.14) can be<br />
shown for the strong version (Result 9.8) as well with not <strong>to</strong>o much difficulties.<br />
Unfortunately, the probabilistic estimate breaks down for the strong form, as we<br />
will discuss below. To make the probabilistic part of the argument work for the<br />
strong case, we have <strong>to</strong> allow more than just one bad l-cube <strong>in</strong>side the L-cubes. In<br />
Section 10.6, we show how <strong>to</strong> deal with this problem.<br />
10.4. Probabilistic estimates - weak form.<br />
We turn <strong>to</strong> the probablistic estimates of the <strong>in</strong>duction step <strong>in</strong> multiscale analysis.<br />
Here, we will prove the multiscale result <strong>in</strong> its weak form (Result 9.6).<br />
In the whole section we assume that the probability distribution P 0 of the <strong>in</strong>dependent,<br />
identically distributed random variables V ω (i) has a bounded density, i. e.<br />
∫<br />
P 0 (A) := P( V ω (i) ∈ A ) = g(λ)dλ,<br />
with ||g|| ∞ = sup | g(λ) | < ∞) . (10.51)<br />
λ<br />
This condition is assumed throughout this section even when not explicitly stated.<br />
The ma<strong>in</strong> result is<br />
THEOREM 10.16. Assume that the probability distribution P 0 has a bounded density.<br />
Suppose L 0 is large enough, γ ≥ 1<br />
L 1/2<br />
, p > 2d and 1 < α <<br />
2p<br />
0 p+2d . If<br />
then for all k<br />
REMARK 10.17.<br />
A<br />
P(Λ L0 is not (2γ, E)−good ) ≤ 1<br />
L 0<br />
p , (10.52)<br />
P(Λ Lk is not (γ, E)−good) ≤ 1<br />
L k<br />
p . (10.53)<br />
• Note that p > 2d ensures that we can choose α > 1.
104<br />
• We need the assumption (10.51) on P 0 (only) <strong>in</strong> order <strong>to</strong> have the Wegner<br />
estimate (Theorem 5.23).<br />
This theorem reduces the multiscale analysis <strong>to</strong> the <strong>in</strong>itial scale estimate (10.52)<br />
which we discuss <strong>in</strong> chapter 11. As we remarked above, Theorem 10.16 is proved<br />
by <strong>in</strong>duction. Thus, under the assumptions of Theorem 10.16 and with the rates γ k<br />
as <strong>in</strong> Theorem 10.14, we have <strong>to</strong> prove the follow<strong>in</strong>g theorem.<br />
THEOREM 10.18. If<br />
P(Λ Lk is not (γ k , E)−good) ≤ 1 L p k<br />
, (10.54)<br />
then<br />
P(Λ Lk+1 is not (γ k+1 , E)−good) ≤ 1<br />
L p . (10.55)<br />
k+1<br />
PROOF: As usual, we set l = L k , L = L k+1 and γ = γ k , γ ′ = γ k+1 . To prove<br />
Theorem 10.18, we use Theorem 10.14 <strong>to</strong> estimate<br />
P ( Λ L is not (γ ′ , E)−good )<br />
≤ P ( Λ L is E−resonant ) (10.56)<br />
+ P ( One of the cubes Λ 2l (m) ⊂ Λ L is E−resonant ) (10.57)<br />
+ P ( There are two disjo<strong>in</strong>t cubes <strong>in</strong> C l (Λ L )<br />
which are not (γ, E)−good ) . (10.58)<br />
Both (10.56) and (10.57) can be bounded us<strong>in</strong>g the Wegner estimate (Theorem<br />
5.23)<br />
provided L is large enough, and<br />
P ( Λ L is E−resonant ) ≤ (2L + 1) d e −√ L<br />
≤ 1 3<br />
1<br />
L p (10.59)<br />
P ( One of the cubes Λ 2l (m) ⊂ Λ L is E−resonant ) (10.60)<br />
≤ (2L + 1) d P ( The cube Λ 2l (0) is E−resonant ) (10.61)<br />
≤ (2L + 1) d (4l + 1) d e −√ 2l<br />
≤ (2L + 1) d (4L 1/α + 1) d e −√ 2 L 2α<br />
1<br />
≤ 1 1<br />
3 L p (10.62)<br />
if L is large enough.<br />
Us<strong>in</strong>g the <strong>in</strong>duction hypothesis (10.54), we can estimate the term (10.58) by
105<br />
∑<br />
i,j∈Λ L<br />
Λ l (i)∩Λ l (j)=∅<br />
≤<br />
∑<br />
P (Λ l (i) and Λ l (j) are both not (γ, E)-good )<br />
i,j∈Λ L<br />
P (Λ l (i) is not (γ, E)-good ) P (Λ l (j) is not (γ, E)-good )<br />
≤ (2L + 1) 2d 1<br />
≤<br />
≤ 1 3<br />
C<br />
L 2p α −2d<br />
provided L is large.<br />
We used above that α <<br />
Summ<strong>in</strong>g up, we get<br />
l 2p<br />
1<br />
L p (10.63)<br />
2p<br />
p+2d implies 2p α<br />
− 2d > p.<br />
P ( Λ L is not (γ ′ , E)−good ) ≤ 1 L p .<br />
□<br />
10.5. Towards the strong form of the multiscale analyis.<br />
When we try <strong>to</strong> prove the ‘uniform’ Result 9.8, i.e. the strong form of the multiscale<br />
estimate, we may proceed <strong>in</strong> the same manner as above for awhile. Let us suppose<br />
we consider two disjo<strong>in</strong>t cubes Λ 1 = Λ L (n) and Λ 2 = Λ L (m). We want <strong>to</strong> prove<br />
We set<br />
P ( For some E ∈ I bothΛ 1 and Λ 2 are not (γ ′ , E)−good ) ≤ L −2p . (10.64)<br />
A 1 (E) = { Λ 1 is not (γ ′ , E)−good }<br />
R 1 (E) = { Λ 1 or a cube <strong>in</strong> C 2l (Λ 1 ) is not E-resonant } (10.65)<br />
B 1 (E) = { C l (Λ 1 ) conta<strong>in</strong>s two disjo<strong>in</strong>t cubes which are not (γ, E)−good } .<br />
We def<strong>in</strong>e A 2 (E), R 2 (E), B 2 (E) analogously for the cube Λ 2 .<br />
The event we are <strong>in</strong>terested <strong>in</strong> (see 10.64) can be expressed through A 1 (E), A 2 (E),<br />
namely<br />
{ ∃ E∈I such that Λ 1 and Λ 2 are not (γ ′ , E)−good } (10.66)<br />
= ⋃ (<br />
A1 (E) ∩ A 2 (E) ) . (10.67)<br />
E∈I<br />
Theorem 10.14 implies that
106<br />
P ( ⋃ E∈I<br />
( A 1 (E) ∩ A 2 (E) ) )<br />
≤<br />
P ( ⋃ E∈I<br />
( R 1 (E) ∩ R 2 (E) ) ) (10.68)<br />
+ P ( ⋃ E∈I<br />
+ P ( ⋃ E∈I<br />
+ P ( ⋃ E∈I<br />
( B 1 (E) ∩ B 2 (E) ) ) (10.69)<br />
( R 1 (E) ∩ B 2 (E) ) ) (10.70)<br />
( B 1 (E) ∩ R 2 (E) ) ) . (10.71)<br />
The term (10.68) can be estimated us<strong>in</strong>g the ‘uniform’ Wegner estimate (Theorem<br />
5.27) and (10.69) will be handled us<strong>in</strong>g the <strong>in</strong>duction hypothesis. It turns out that<br />
the critical terms are the mixed ones (10.70) and (10.71).<br />
The only effective way we know <strong>to</strong> estimate (10.71) is<br />
P ( ⋃ E∈I<br />
( B 1 (E) ∩ R 2 (E) ) ) ≤ P ( ⋃ E∈I<br />
B 1 (E) )<br />
≤ L 2d 1<br />
≤<br />
l 2p<br />
1<br />
L 2p/α−2d (10.72)<br />
where we used the <strong>in</strong>duction hypothesis and the fact that there are at most L 2d<br />
disjo<strong>in</strong>t cubes of side length l <strong>in</strong> Λ 1 .<br />
Observe that the term ⋃ E∈I R 2(E) which we neglected above does not have small<br />
probability as long as there is spectrum <strong>in</strong>side I.<br />
S<strong>in</strong>ce we need α > 1, the exponent <strong>in</strong> (10.72) is certa<strong>in</strong>ly less than 2p. Consequently<br />
there is no way <strong>to</strong> do the <strong>in</strong>duction step the way we tried above. The<br />
<strong>in</strong>duction step would require that (10.72) is less than 1 .<br />
L 2p<br />
Observe that the situation is completely analogous <strong>to</strong> the one <strong>in</strong> Section 10.2 (see<br />
(10.38)). There we needed <strong>to</strong> allow more (namely one) bad cubes. The same idea<br />
remedies the present situation: We have <strong>to</strong> accept ‘three’ bad cubes, as will be<br />
expla<strong>in</strong>ed <strong>in</strong> the next section.<br />
10.6. Estimates - third try .<br />
In a third round, we accept ‘three’ bad cubes. More precisely: We assume that the<br />
cube Λ L does not conta<strong>in</strong> four disjo<strong>in</strong>t cubes of side length l which are not (γ, E)-<br />
good. Then, there are (at most) three cubes, Λ 2l (m 1 ), Λ 2l (m 2 ), Λ 2l (m 3 ) ⊂ Λ L ,<br />
such that there are no bad cubes outside M = ⋃ 3<br />
ν=1 Λ 2l(m ν ).<br />
As <strong>in</strong> Section 10.3, we use the geometric resolvent equation and an exponential<br />
bound on the Green’s function as long as we do not enter one of the Λ 2l (m ν ).<br />
Once we enter such a set, we would like <strong>to</strong> use the geometric resolvent equation <strong>in</strong>
connection with a Wegner-type bound for Λ 2l (m ν ) as <strong>in</strong> the follow<strong>in</strong>g expression<br />
for u ∈ Λ 2l (m ν ):<br />
| G Λ L<br />
E (u, n)| ≤ ∑<br />
(q,q ′ )∈∂Λ 2l (mν )<br />
q∈ Λ 2l (mν )<br />
107<br />
| G Λ 2l(m ν)<br />
E<br />
(u, q)| | G Λ L<br />
E (q′ , n)| . (10.73)<br />
If we assume that Λ 2l (m ν ) is not E-resonant we can estimate the first term on the<br />
right hand side of (10.73) by e √ 2l . If the site q ′ is the center of a good cube, we may<br />
estimate the second term G Λ L<br />
E (q′ , n) by apply<strong>in</strong>g the geometric resolvent equation<br />
for the cube Λ l (q ′ ) and us<strong>in</strong>g the exponential bound for this cube. However, it<br />
is not guaranteed that Λ l (q ′ ) is (γ, E)-good. q ′ could belong <strong>to</strong> one of the other<br />
‘dangerous’ cubes Λ 2l (m ν ′). The problem here is that two (or all three) of these<br />
cubes could <strong>to</strong>uch or <strong>in</strong>tersect.<br />
To get rid of this problem, we redef<strong>in</strong>e the ‘dangerous’ regions where we use the<br />
Wegner bound <strong>in</strong>stead of the exponential bound. We say that two subsets A and<br />
B of Z d <strong>to</strong>uch if A ∩ B ≠ ∅ or if there are po<strong>in</strong>ts x ∈ A and y ∈ B such that<br />
||x − y|| ∞ = 1.<br />
As before we use the geometric resolvent equation iteratively <strong>to</strong> estimate the Green’s<br />
function G Λ L<br />
E . We def<strong>in</strong>e sets M 1, M 2 , M 3 - the dangerous regions - where we use<br />
the Wegner estimate, i. e. we will assume that the M i are not E-resonant. We<br />
construct the M i <strong>in</strong> such a way that for all sites x outside the M i , the cube Λ l (x) is<br />
(γ, E)-good. Moreover, any two of the M i do not <strong>to</strong>uch.<br />
If the cubes Λ 2l (m ν ) do not <strong>to</strong>uch each other, we set M ν = Λ 2l (m ν ).<br />
If two of the Λ 2l (m ν ) <strong>to</strong>uch, say Λ 2l (m 1 ) and Λ 2l (m 2 ), we set M ′ = Λ 6l+1 (m 1 ).<br />
Then Λ 2l (m 1 ) ∪ Λ 2l (m 2 ) ⊂ M ′ . Indeed, if Λ 2l (m 1 ) and Λ 2l (m 2 ) <strong>to</strong>uch, there are<br />
po<strong>in</strong>ts x ∈ Λ 2l (m 1 ) and y ∈ Λ 2l (m 2 ) with || x − y|| ∞ ≤ 1. If z ∈ Λ 2l (m 2 ) we<br />
have<br />
||z − m 1 || ∞<br />
≤ ||z − m 2 || ∞ + ||m 2 − y|| ∞ + ||y − x|| ∞ + ||x − m 1 || ∞<br />
≤ 6 l + 1 . (10.74)<br />
If M ′ and Λ 2l (m 3 ) do not <strong>to</strong>uch we set M 1 = M ′ and M 2 = Λ 2l (m 3 ) (The set M 3<br />
is not needed, we may formally set M 3 = ∅.). If M ′ and Λ 2l (M 3 ) do <strong>to</strong>uch then<br />
M ′ , Λ 2l (m 3 ) ⊂ Λ 10l+2 (m 1 ) which is shown by a calculation analogous <strong>to</strong> (10.74).<br />
In this case, we set M 1 = Λ 10l+2 (m 1 ) and M 2 = M 3 = ∅.<br />
We have shown<br />
LEMMA 10.19. If there are not four disjo<strong>in</strong>t cubes <strong>in</strong> C l (Λ L ) which are not (γ, E)-<br />
good, then either<br />
• There are three cubes M 1 , M 2 , M 3 ∈ C 2l (Λ L ) which do not <strong>to</strong>uch and<br />
such that any cube <strong>in</strong> C l (Λ L ) with center outside the M i is (γ, E)-good,<br />
or
108<br />
• There is a cube M 1 ∈ C 6l+1 (Λ L ) and a cube M 2 ∈ C 2l (Λ L ) which do not<br />
<strong>to</strong>uch such that any cube <strong>in</strong> C l (Λ L ) with center outside the M i is (γ, E)-<br />
good,<br />
or<br />
• There is a cube M 1 ∈ C 10l+2 (Λ L ) such that any cube <strong>in</strong> C l (Λ L ) with<br />
center outside M 1 is (γ, E)-good.<br />
We are now <strong>in</strong> a position <strong>to</strong> prove the analytic part of the <strong>in</strong>duction step of multiscale<br />
analysis <strong>in</strong> the f<strong>in</strong>al form.<br />
THEOREM 10.20. Suppose L 0 is large enough and L k+1 = L k α with 1 < α < 2.<br />
If for a certa<strong>in</strong> k (l := L k and L := L k+1 )<br />
(1) there do not exist four disjo<strong>in</strong>t cubes <strong>in</strong> C l (Λ L ) which are not (γ k , E)-<br />
good with a rate γ k ≥ 12 ,<br />
l<br />
(2) no cube <strong>in</strong><br />
1/2<br />
C 2l (Λ L ) ∪ C 6l+1 (Λ L ) ∪ C 10l+2 (Λ L ) (10.75)<br />
is E-resonant and<br />
(3) the cube Λ L is not E-resonant,<br />
then the cube Λ L is (γ k+1 , E)- good with a rate γ k+1 satisfy<strong>in</strong>g γ k+1 ≥ 12<br />
L 1/2 .<br />
Moreover we can choose the rate γ k+1 such that<br />
γ k+1 ≥ γ k − γ k<br />
C<br />
L k<br />
α−1 − C<br />
L k<br />
α/2 . (10.76)<br />
As above, we can estimate the decay rates as follows.<br />
COROLLARY 10.21. If L 0 is large enough and the <strong>in</strong>itial rate γ 0 satisfies γ 0 ≥<br />
12<br />
L 1/2<br />
then the γ k <strong>in</strong> Theorem 10.20 satisfy γ k ≥ γ 0<br />
0 2<br />
for all k.<br />
PROOF: We set γ = γ k and γ ′ = γ k+1 . From Lemma 10.19 we know that there<br />
are three cubes M 1 , M 2 , M 3 of side length 2l, 6l + 1 or 10l + 2 (or 0 if M i = ∅)<br />
such that the M i conta<strong>in</strong> all cubes <strong>in</strong> C l (Λ L ) which are not (γ, E)-good.<br />
Start<strong>in</strong>g with m ∈ Λ √ L and n ∈ ∂− Λ L , we use the geometric resolvent equation<br />
repeatedly.<br />
If m does not belong <strong>to</strong> one of the ‘dangerous’ cubes M i we know Λ l (m) is (γ, E)-<br />
good, so we estimate<br />
| G Λ L<br />
E (m, n)| ≤ ∑<br />
(q,q ′ )∈∂Λ l (m)<br />
q∈ Λ l (m)<br />
| G Λ l(m)<br />
E<br />
(m, q)| | G Λ L<br />
E (q′ , n)| (10.77)<br />
≤ 2d (2l + 1) d−1 e −γl | G Λ L<br />
E (n 1, n)| (10.78)<br />
≤ e −˜γl | G Λ L<br />
E (n 1, n)| . (10.79)<br />
We call such a step an exponential bound. We do this repeatedly, as long as the<br />
new po<strong>in</strong>t n 1 , n 2 , . . . neither belongs <strong>to</strong> one of the M i nor is close <strong>to</strong> the boundary<br />
of Λ L .
109<br />
If n j belongs <strong>to</strong> one of the M i , say <strong>to</strong> M 1 , we use a Wegner-type bound<br />
| G Λ L<br />
E (n j, n)|<br />
≤<br />
≤<br />
∑<br />
| G M 1<br />
E<br />
(n j, q)| | G Λ L<br />
E (q′ , n)|<br />
(q,q ′ )∈∂M 1<br />
q∈ M 1<br />
2d (20l + 5) d−1 e √ 10l+2 | G Λ L<br />
E (n′ j, n)| (10.80)<br />
for a certa<strong>in</strong> n ′ j ∈ ∂+ M 1 . S<strong>in</strong>ce the M i do not <strong>to</strong>uch, we can be sure that Λ l (n ′ j ) is<br />
(γ, E)-good. Consequently, we can always (as long as n ′ j is not near the boundary<br />
of Λ L ) do an exponential bound after a Wegner-type bound and obta<strong>in</strong><br />
| G Λ L<br />
E (n j, n)|<br />
≤ 2d (20l + 5) d−1 e √ 10l+2 | G Λ L<br />
E (n′ j, n)|<br />
≤ (2d) 2 (20l + 5) d−1 (2l + 1) d−1 e √ 10l+2 e −γ l | G Λ L<br />
E (n j+1, n)| . (10.81)<br />
If l is larger than a certa<strong>in</strong> constant and γ ≥ 12<br />
l 1/2 , we have<br />
thus<br />
ρ = (2d) 2 (20l + 5) d−1 (2l + 1) d−1 e √ 10l+2 e −γ l ≤ 1 (10.82)<br />
(10.81) ≤ | G Λ L<br />
E (n j+1, n)| . (10.83)<br />
Whenever the po<strong>in</strong>t n j does not belong <strong>to</strong> one of the ‘dangerous’ regions M i ,<br />
we know that Λ l (n j ) is (γ, E)-good. Hence, we obta<strong>in</strong> an exponential bound of<br />
the Green’s function and ga<strong>in</strong> an exponential fac<strong>to</strong>r e −γ l . This step can be done<br />
roughly L l<br />
times. Hence, we get the desired bound. The details are as <strong>in</strong> the previous<br />
sections.<br />
□<br />
Now, we do the probabilistic estimate.<br />
THEOREM 10.22. Assume that the probability distribution P 0 has a bounded density.<br />
Suppose L 0 is large enough, γ ≥ 12<br />
L 1/2<br />
, p > 2d and 1 < α <<br />
2p<br />
0 p+2d<br />
. If for<br />
any disjo<strong>in</strong>t cubes Λ L0 (n) and Λ L0 (m)<br />
P (<br />
For some E ∈ I both Λ L0 (n) and Λ L0 (m)<br />
are not (2γ, E)−good ) 1<br />
(10.84)<br />
≤<br />
2p<br />
L 0<br />
then for all k and all disjo<strong>in</strong>t cubes Λ Lk (n) and Λ Lk (m)<br />
P (<br />
For some E ∈ I both Λ Lk (n) and Λ Lk (m)<br />
are not (γ, E)−good ) 1<br />
≤<br />
2p<br />
L . (10.85)<br />
k
110<br />
PROOF: The prove works by <strong>in</strong>duction. So, we suppose, we know (10.85) already<br />
for k. We try <strong>to</strong> prove it for k + 1.<br />
As usual, we set l = L k , L = L k+1 , γ = γ k , and γ ′ = γ k+1 .<br />
We also abbreviate Λ 1 = Λ Lk+1 (n) and Λ 2 = Λ Lk+1 (m).<br />
Similar <strong>to</strong> (10.65) we def<strong>in</strong>e (i = 1, 2)<br />
A i (E) = { Λ i is not (γ ′ , E)−good }<br />
Q i (E) = { Λ i or a cube <strong>in</strong> C 2l (Λ i ) ∪ C 6l+1 (Λ i ) ∪ C 10l+2 (Λ i ) is not E-resonant }<br />
D i (E) = { C l (Λ i ) conta<strong>in</strong>s four disjo<strong>in</strong>t cubes which are not (γ, E)−good } .<br />
Let us denote by S(E) the set of all cubes of side length l which are not (γ, E)-<br />
good. Like <strong>in</strong> Section 10.5, we estimate<br />
P ( ∃ E∈I such that Λ 1 and Λ 2 are not (γ ′ , E)−good )<br />
( ⋃ (<br />
= P A1 (E) ∩ A 2 (E) ) )<br />
E∈I<br />
( ⋃<br />
) ( ⋃<br />
)<br />
≤ P ( Q 1 (E) ∩ Q 2 (E) ) + P ( D 1 (E) ∩ D 2 (E) )<br />
E∈I<br />
E∈I<br />
≤<br />
≤<br />
( ⋃<br />
)<br />
+ P ( Q 1 (E) ∩ D 2 (E) )<br />
E∈I<br />
( ⋃<br />
)<br />
P ( Q 1 (E) ∩ Q 2 (E) )<br />
E∈I<br />
)<br />
( ⋃<br />
+ P<br />
E∈I<br />
D 1 (E)<br />
( ⋃<br />
+ P<br />
E∈I<br />
( ⋃<br />
)<br />
P ( Q 1 (E) ∩ Q 2 (E) )<br />
E∈I<br />
( ⋃<br />
)<br />
+ P ( D 1 (E) ∩ Q 2 (E) )<br />
E∈I<br />
)<br />
( ⋃<br />
+ P<br />
E∈I<br />
)<br />
D 2 (E)<br />
D 1 (E)<br />
( ⋃<br />
+ 3 P<br />
E∈I<br />
( ⋃<br />
P<br />
E∈I<br />
)<br />
D 1 (E) .<br />
)<br />
D 2 (E)<br />
Let us first estimate the latter term:<br />
( ⋃<br />
P<br />
E∈I<br />
= P ( ∃<br />
E ∈ I<br />
≤<br />
∑<br />
)<br />
D 1 (E)<br />
C i ∈ C l (Λ L )<br />
pairwise disjo<strong>in</strong>t<br />
∃<br />
C 1 , C 2 , C 3 , C 4 ∈ S(E) )<br />
C 1 , C 2 , C 3 , C 4 ∈ C l (Λ L )<br />
pairwise disjo<strong>in</strong>t<br />
P (<br />
∃<br />
E ∈ I C 1 ∈ S(E), C 2 ∈ S(E), C 3 ∈ S(E) and C 4 ∈ S(E) )
≤<br />
∑<br />
C i ∈ C l (Λ L )<br />
pairwise disjo<strong>in</strong>t<br />
P<br />
( (<br />
∃<br />
E ∈ I<br />
(<br />
∃<br />
E ∈ I<br />
)<br />
C 1 ∈ S(E) and C 2 ∈ S(E)<br />
) )<br />
C 3 ∈ S(E) and C 4 ∈ S(E)<br />
and<br />
111<br />
≤<br />
∑<br />
C i ∈ C l (Λ L )<br />
pairwise disjo<strong>in</strong>t<br />
≤ C L 4d ( 1<br />
l 2p ) 2<br />
(<br />
P<br />
≤<br />
)<br />
∃<br />
E ∈ I C 1, C 2 ∈ S(E)<br />
C<br />
L 4p/α − 4d ≤ 1 4<br />
1<br />
L 2p .<br />
2p<br />
p+2d .<br />
In the last step, we used that p > 2d and 1 < α <<br />
We turn <strong>to</strong> the estimate of<br />
( ⋃<br />
)<br />
P ( Q 1 (E) ∩ Q 2 (E) )<br />
E∈I<br />
(<br />
P ∃<br />
E ∈ I C 3, C 4 ∈ S(E)<br />
)<br />
By sett<strong>in</strong>g Q i = C 2l (Λ i ) ∪ C 6l+1 (Λ i ) ∪ C 10l+2 (Λ i ) ∪ {Λ i }, we get<br />
( ⋃<br />
)<br />
∑ (<br />
P ( Q 1 (E)∩Q 2 (E) ) ≤<br />
P<br />
E∈I<br />
c 1 ∈Q 1 ,c 2 ∈Q 2<br />
∃ c 1 and c 2 are E-resonant<br />
E∈I<br />
(10.86)<br />
)<br />
.<br />
Each term <strong>in</strong> the sum <strong>in</strong> (10.86) can be estimated us<strong>in</strong>g Theorem 5.27 by a term of<br />
the form C L k e L 1 2<br />
and the sum does not have more than C L d terms, thus the sum<br />
can certa<strong>in</strong>ly be bounded by 1 1<br />
4<br />
.<br />
L 2p<br />
This f<strong>in</strong>ishes the proof.<br />
□<br />
Notes and Remarks<br />
The celebrated paper by Fröhlich and Spencer [47] laid the foundation for multiscale<br />
analysis. This technique was further developed and substantially simplified<br />
<strong>in</strong> the paper by Dreifus and Kle<strong>in</strong> [40]. Germ<strong>in</strong>et and Kle<strong>in</strong> [49] developed a<br />
‘Bootstrap multiscale analysis’ which uses the output of a multiscale estimate as<br />
the <strong>in</strong>put of a new multiscale procedure. These authors obta<strong>in</strong> the best available<br />
estimates of this k<strong>in</strong>d. In fact, <strong>in</strong> [50] they prove that their result characterizes the<br />
regime of ‘strong localization’.<br />
Multiscale analysis can be transferred <strong>to</strong> the cont<strong>in</strong>uous case as well, see e.g. [100,<br />
24, 11, 59, 80, 44, 70, 71, 137].
113<br />
11.1. Large disorder.<br />
11. The <strong>in</strong>itial scale estimate<br />
In this f<strong>in</strong>al chapter, we will prove an <strong>in</strong>itial scale estimate for two cases, namely for<br />
energies near the bot<strong>to</strong>m of the spectrum with arbitrary disorder and for arbitrary<br />
energies at large disorder.<br />
We prove the <strong>in</strong>itial scale estimate first for the case of high disorder. As usual we<br />
have <strong>to</strong> assume that the random variables V ω (n) are <strong>in</strong>dependent and identically<br />
distributed with a bounded density g(λ). We may say that the disorder is high if<br />
the norm ‖ g ‖ ∞ is small. In fact, small ‖ g ‖ ∞ reflects a wide spread<strong>in</strong>g of the<br />
random variables.<br />
THEOREM 11.1. Suppose the distribution P 0 has a bounded density g.<br />
Then for any L 0 and any γ > 0, there is a ρ > 0 such that:<br />
If ‖ g ‖ ∞ < ρ and Λ 1 = Λ L0 (n), Λ 2 = Λ L0 (m) are disjo<strong>in</strong>t, then<br />
PROOF:<br />
P ( ∃ E Λ 1 and Λ 2 are both not (γ, E)-good ) ≤ 1<br />
L 0<br />
2p . (11.1)<br />
S<strong>in</strong>ce | G Λ i<br />
E (m, n)| ≤ ‖ (H Λ i<br />
− E) −1 ‖ we have<br />
P ( ∃ E Λ 1 and Λ 2 are both not (γ, E)-good )<br />
≤ P ( ∃ E ‖ (H Λ1 − E) −1 ‖ > e −γ L 0<br />
and ‖ (H Λ2 − E) −1 ‖ > e −γ L 0<br />
)<br />
≤ P ( ∃ E dist(E, σ(H Λ1 ) ≤ e γ L 0<br />
and dist(E, σ(H Λ2 ) ≤ e γ L 0<br />
)<br />
≤<br />
2 C ‖ g ‖ ∞ e γ L 0<br />
(2 L 0 + 1) 2d<br />
where we used the ‘uniform’ Wegner estimate, Theorem 5.27, <strong>in</strong> the f<strong>in</strong>al estimate.<br />
By choos<strong>in</strong>g ρ and, hence, ‖ g ‖ ∞ very small we obta<strong>in</strong> the desired estimate. □<br />
11.2. The Combes-Thomas estimate.<br />
To prove the <strong>in</strong>itial scale estimate for small energies, the follow<strong>in</strong>g bound is crucial.<br />
THEOREM 11.2 (Combes-Thomas estimate). If H = H 0 +V is a discrete Schröd<strong>in</strong>ger<br />
opera<strong>to</strong>r on l 2 (Z d ) and dist(E, σ(H)) = δ ≤ 1, then for any n, m ∈ Z d<br />
∣<br />
∣(H − E) −1 (n, m) ∣ ∣ ≤ 2 δ<br />
e −<br />
δ<br />
12 d || n−m|| 1<br />
. (11.2)<br />
REMARK 11.3. Theorem 11.2 can be improved <strong>in</strong> various directions, see for example<br />
the discussion of the Combes-Thomas estimate <strong>in</strong> [128]. In particular, the<br />
condition δ ≤ 1 which we need for technical reasons is rather unnatural. Our proof<br />
δ<br />
can easily be extended <strong>to</strong> δ ≤ C for any C < ∞ but then the exponent<br />
12 d<br />
<strong>in</strong> the<br />
right hand side of (11.2) has <strong>to</strong> be adjusted depend<strong>in</strong>g on the value of C.<br />
PROOF: For fixed µ > 0 <strong>to</strong> be specified later and fixed n 0 , we def<strong>in</strong>e the multiplication<br />
opera<strong>to</strong>r F = F n0 on l 2 (Z d ) by
114<br />
F u(n) = F n0 u(n) = e µ || n 0−n|| 1<br />
u(n) . (11.3)<br />
Then for any opera<strong>to</strong>r A we have<br />
( )<br />
Fn −1<br />
0<br />
A F n0 (n, m) = e −µ || n 0−n|| 1<br />
A(n, m) e µ || n 0−m|| 1<br />
. (11.4)<br />
Hence, with F = F n<br />
∣ (H − E) −1 (n, m) ∣ ∣ = e − µ || n−m|| 1 ∣ ∣ F −1 (H − E) −1 F (n, m) ∣ ∣<br />
= e − µ || n−m|| 1 ∣ ∣ (F −1 H F − E) −1 (n, m) ∣ ∣<br />
≤ e − µ || n−m|| 1<br />
‖ (F −1 H F − E) −1 ‖ . (11.5)<br />
To compute the norm of the opera<strong>to</strong>r (F −1 H F − E) −1 , we use the resolvent<br />
equation <strong>to</strong> conclude<br />
This implies<br />
(F −1 H F − E) −1<br />
= (H − E) −1 − (F −1 H F − E) −1( F −1 HF − H ) (H − E) −1 .<br />
(F −1 H F − E) −1 (1 + (F −1 HF − H)(H − E) −1 ) = (H − E) −1 .<br />
If ‖(F −1 HF −H)(H−E) −1 ‖ ≤ 1, we may <strong>in</strong>vert (1+(F −1 HF −H)(H−E) −1 )<br />
and obta<strong>in</strong><br />
(F −1 HF − E) −1<br />
= (H − E) −1( 1 + (F −1 HF − H)(H − E) −1) −1<br />
. (11.6)<br />
We compute the norm of the opera<strong>to</strong>r F −1 HF − H. If an opera<strong>to</strong>r A on l 2 (Z d )<br />
has matrix elements A(u, v), then A is bounded if<br />
∑ ∣<br />
a 1 = sup<br />
A(u, v) ∣ < ∞ and<br />
u∈ Z d v∈ Z<br />
∑<br />
d ∣ ∣<br />
a 2 = sup ∣A(u, v) < ∞ .<br />
v∈ Z d u∈ Z d<br />
Moreover, we have<br />
‖ A ‖ ≤ a 1 1/2 a 2<br />
1/2<br />
(see e.g. [141]). We estimate us<strong>in</strong>g (11.4)<br />
(11.7)<br />
∑<br />
v∈ Z d<br />
| ( F −1<br />
n HF n − H ) (u, v) | ≤<br />
∑<br />
v:|| v−u|| 1 =1<br />
|| e −µ|| n−u|| 1<br />
e µ|| n−v|| 1<br />
− 1||<br />
The last <strong>in</strong>equality results from an elementary calculation:<br />
≤ 2 d µ e µ . (11.8)
115<br />
For || u − v|| 1 ≤ 1 we have<br />
∣ || n − u|| 1 − || n − v|| 1<br />
∣ ∣ ≤ || u − v|| 1 ≤ 1 .<br />
Moreover, for |a| ≤ 1 and µ > 0<br />
∣ e µa − 1 ∣ ≤ ∣ e µ − 1 ∣ = e µ − 1<br />
≤<br />
∫ 1<br />
0<br />
µ e µ t ≤ µ e µ<br />
which proves (11.8).<br />
Estimate (11.8) and an analogous estimate with the role of u and v <strong>in</strong>terchanged<br />
imply us<strong>in</strong>g (11.7)<br />
Now we choose µ =<br />
‖ F −1 HF − H ‖ ≤ 2 d µ e µ . (11.9)<br />
δ<br />
12 d<br />
. As dist(E, σ(H)) = δ ≤ 1 we conclude<br />
‖(F −1 HF − H)(H − E) −1 ‖ ≤ ‖(F −1 HF − H)‖ ‖(H − E) −1 ‖<br />
≤ 2 d µ e µ 1 δ<br />
δ<br />
= 2 d<br />
12 d e δ<br />
12 d<br />
≤ 1 2<br />
1<br />
δ<br />
(11.10)<br />
Above we used e δ<br />
12 d ≤ e ≤ 3 s<strong>in</strong>ce δ ≤ 1.<br />
It follows that the opera<strong>to</strong>r 1 + (F −1 HF − H)(H − E) −1 is <strong>in</strong>deed <strong>in</strong>vertible and,<br />
us<strong>in</strong>g the Neumann series, we conclude that<br />
(<br />
∣∣<br />
1 + (F −1 HF − H)(H − E) −1) −1 ∣ ∣ ∣∣ ∣ ≤ 2 (11.11)<br />
Thus, by (11.6) we have<br />
∣ ∣ ∣ (F −1 HF − E) −1∣ ∣ ∣ ∣ =<br />
∣ ∣<br />
∣ ∣(H − E)<br />
−1 ( 1 + (F −1 HF − H) (H − E) −1) −1 ∣ ∣ ∣ ∣<br />
≤ 2 δ<br />
(11.12)<br />
and (11.5) gives<br />
∣ (H − E) −1 (n, m) ∣ ∣ ≤ e − µ || n−m|| 1<br />
‖ (F −1 H F − E) −1 ‖<br />
≤ 2 δ e−<br />
δ<br />
12 d || n−m|| 1<br />
. (11.13)<br />
□
116<br />
11.3. Energies near the bot<strong>to</strong>m of the spectrum.<br />
For energies near the bot<strong>to</strong>m of the spectrum, we prove the follow<strong>in</strong>g estimate.<br />
THEOREM 11.4. Suppose the distribution P 0 has a bounded support. Denote by<br />
E 0 the <strong>in</strong>fimum of the spectrum of H ω . Then for arbitrary large L 0 , any C and p<br />
there is an energy E 1 > E 0 such that<br />
C<br />
P ( Λ L0 is not (<br />
1/2<br />
L , E)-regular for some E ≤ E 1 ) ≤ 1<br />
p<br />
0<br />
L . (11.14)<br />
0<br />
REMARK 11.5. By the results of Chapter 10 the above result implies pure po<strong>in</strong>t<br />
spectrum for energies near the bot<strong>to</strong>m of the spectrum.<br />
PROOF: If E 0 (H ΛL ) ≥ 2γ then Theorem 11.2 implies that Λ L is (γ, E)-regular<br />
for any E ≤ γ. Indeed, for such an E<br />
( )<br />
dist E, σ(H ΛL ) ≥ E 0 (H ΛL ) − γ ≥ γ . (11.15)<br />
From our study of Lifshitz tails (Chapter 6), we have already a lower bound on<br />
some E 0 (HΛ N L<br />
) ≤ E 0 (H ΛL ) , namely:<br />
By (6.10) and Lemma 6.4 there exist l 0 and β such that<br />
(<br />
P<br />
E 0 (H N Λ l0<br />
) < 1<br />
β l 0<br />
2<br />
)<br />
≤ e −c l 0 d . (11.16)<br />
1<br />
This estimate tells us that for E ≤ γ = 2 , the cube Λ<br />
2 β l l0 is (γ, E)-good with<br />
0<br />
very high probability.<br />
This sounds like it is exactly what we need for the <strong>in</strong>itial scale estimate. Unfortunately,<br />
it is not quite what makes the mach<strong>in</strong>e work.<br />
The multiscale scheme requires for the <strong>in</strong>itial step the assumption (see Theorem<br />
10.22)<br />
γ ≥<br />
C<br />
(11.17)<br />
L 0<br />
1/2<br />
but the γ we obta<strong>in</strong> from (11.16) is much smaller than the rate required by (11.17).<br />
On the other hand, the right hand side of (11.16) is much better than what we need<br />
(exponential versus polynomial bound). So, we may hope we can ‘trade probability<br />
for rate’. This is exactly what we do now.<br />
We build a big cube Λ L0 by pil<strong>in</strong>g up disjo<strong>in</strong>t copies of the cube Λ l0 , more precisely<br />
Λ L0<br />
= ⋃<br />
j∈R<br />
Λ l0 (j) . (11.18)<br />
Indeed, for any odd <strong>in</strong>teger r we may take L 0 = r l 0 + r−1<br />
2<br />
. The set R <strong>in</strong> (11.18)<br />
conta<strong>in</strong>s r d po<strong>in</strong>ts.<br />
By (5.50) we have<br />
HΛ N L0<br />
≥ ⊕ HΛ N l0<br />
(j) , (11.19)<br />
j∈R
117<br />
hence<br />
It follows that<br />
E 0<br />
(<br />
H N Λ L0<br />
)<br />
( )<br />
≥ <strong>in</strong>f E 0 HΛ N j∈R l0<br />
(j) . (11.20)<br />
( ) )<br />
P E 0<br />
(HΛ N L0<br />
≤ 2 γ<br />
( ( ) )<br />
≤ P <strong>in</strong>f E 0 HΛ N j∈R l0<br />
(j) ≤ 2 γ<br />
( ( )<br />
)<br />
≤ P E 0 HΛ N l0<br />
(j) ≤ 2 γ for some j ∈ R<br />
( ) )<br />
≤ r d P E 0<br />
(HΛ N l0<br />
≤ 2 γ . (11.21)<br />
1<br />
If we choose γ = 2 , we may use (11.16) <strong>to</strong> estimate (11.21) and obta<strong>in</strong><br />
2 β l 0<br />
( ) )<br />
P E 0<br />
(HΛ N L0<br />
≤ 2 γ ≤ r d e − c l 0 d . (11.22)<br />
Now, we choose r and hence L 0 <strong>in</strong> such a way that γ ><br />
C<br />
L 1/2<br />
. This leads <strong>to</strong> sett<strong>in</strong>g<br />
0<br />
r ∼ l 3 0 , thus L 0 ∼ l 4 0 . With this choice, (11.22) gives<br />
( ) )<br />
P E 0<br />
(HΛ N L0<br />
≤ 2 γ ≤ C 1 L d 0 e − c′ L d/4 0<br />
. (11.23)<br />
S<strong>in</strong>ce the right hand side of (11.23) is smaller than 1<br />
L 0<br />
p , this proves the <strong>in</strong>itial scale<br />
estimate.<br />
□<br />
Notes and Remarks<br />
Already the paper [47] conta<strong>in</strong>ed the proof for high disorder localization we gave<br />
above. The idea <strong>to</strong> use Lifshitz tails <strong>to</strong> prove localization for small energies goes<br />
back <strong>to</strong> [100] and was further developed <strong>in</strong> [70] (see also [73]), but an <strong>in</strong>timate<br />
connection between Lifshitz tails and <strong>An</strong>derson localization was clear <strong>to</strong> physicists<br />
for a long time (see [98]).<br />
The Combes-Thomas <strong>in</strong>equality was proved <strong>in</strong> [28]. It was improved <strong>in</strong> [11], see<br />
also [128]. We <strong>to</strong>ok the proof above from [1].
119<br />
12. Appendix: Lost <strong>in</strong> Multiscalization – A guide through the jungle<br />
This is a short guide <strong>to</strong> the proof of <strong>An</strong>derson localization via multiscale analysis<br />
given <strong>in</strong> this text.<br />
The core of the localization proof is formed by the estimates stated <strong>in</strong> Section 9.3<br />
as Result 9.6 and 9.8. The first estimate (9.6) says that for a given energy E, exponential<br />
decay of the Green’s function is very likely on large cubes. Cubes with<br />
exponentially decay<strong>in</strong>g Green’s functions will be called ‘good’ cubes. In Section<br />
9.2 we prove that the estimate <strong>in</strong> Result 9.6 implies the absence of absolutely cont<strong>in</strong>uous<br />
spectrum.<br />
The strong version (Result 9.8) of the multiscale estimate considers a whole energy<br />
<strong>in</strong>terval I and two disjo<strong>in</strong>t cubes. The result tells us that with high probability for<br />
all energies <strong>in</strong> I at least one of the cubes has an exponentially decay<strong>in</strong>g resolvent.<br />
This result is a strong version of the former result as it is uniform <strong>in</strong> the energy. The<br />
price <strong>to</strong> be paid is the consideration of a second cube. A s<strong>in</strong>gle cube cannot be good<br />
for all energies <strong>in</strong> I if there is spectrum at all <strong>in</strong> I (see 9.2). We show <strong>in</strong> Section 9.5<br />
that the strong form of the multiscale estimate implies pure po<strong>in</strong>t spectrum <strong>in</strong>side<br />
I. This is done us<strong>in</strong>g the exponential decay of eigenfunctions which we deduce<br />
from the key Theorem 9.9. The connection between spectrum and (generalized)<br />
eigenfunctions is discussed <strong>in</strong> Chapter 7.<br />
The proofs of the multiscale estimates (Results 9.6 and 9.8) are conta<strong>in</strong>ed <strong>in</strong> the<br />
Chapters 10 and 11. We prove the estimates <strong>in</strong>ductively for cubes of side length<br />
L k , k = 0, 1, 2 . . . . The length scale is such that L k+1 = L α k<br />
for an α > 1.<br />
The <strong>in</strong>duction step from L k <strong>to</strong> L k+1 is done <strong>in</strong> Chapter 10. In a first attempt<br />
(Section 10.2) <strong>to</strong> do this for the weaker form we prove that if all the small cubes<br />
(of size L k ) <strong>in</strong>side a big cube (of size L k+1 ) are good, then the big cube itself is<br />
good if we have a rough a priori estimate for the big cube. This a priori bound is<br />
provided by the ‘Wegner estimate’, a key <strong>in</strong>gredient <strong>to</strong> our proof. We prove the<br />
Wegner estimate <strong>in</strong> Section 5.5. Unfortunately, the probability that all small cubes<br />
<strong>in</strong>side the big one are good is rather small. So, this ‘first try’ is not appropriate <strong>to</strong><br />
prove that the big cube is good with high enough probability.<br />
In the ‘second try’ we allow one bad small cube <strong>in</strong>side the big cube. (For the<br />
precise formulation see Section 10.3). To prove that this still implies that the big<br />
cube is good requires more work. We need aga<strong>in</strong> that the big cube and also the<br />
‘bad’ small cube allow an a priori bound of the Wegner type. The advantage of<br />
allow<strong>in</strong>g one bad cube is that this event has a much higher probability. In this way,<br />
we prove the <strong>in</strong>duction step for the weak form of the multiscale analysis.<br />
The strong form of the multiscale analysis is then treated <strong>in</strong> Section 10.6. Here we<br />
have <strong>to</strong> allow even a few bad cubes among the small ones. This makes the proof<br />
yet a bit more complicated.<br />
So far we have done the <strong>in</strong>duction step. Of course, we still have <strong>to</strong> prove the<br />
estimate for the <strong>in</strong>itial length L 0 . This is done <strong>in</strong> Chapter 11. We prove that the<br />
<strong>in</strong>itial estimate is satisfied if either the disorder is large or the energy is close <strong>to</strong> the<br />
bot<strong>to</strong>m of the spectrum. <strong>An</strong> important <strong>to</strong>ol <strong>in</strong> this chapter is the Combes-Thomas<br />
<strong>in</strong>equality. We prove this result <strong>in</strong> section 11.2.
120<br />
The strategy of proof outl<strong>in</strong>ed above is certa<strong>in</strong>ly not the fastest one <strong>to</strong> prove localization<br />
via multiscale analysis. However, we believe that for a first read<strong>in</strong>g, it is<br />
easier <strong>to</strong> learn the subject this way than <strong>in</strong> a streaml<strong>in</strong>ed turbo version.
121<br />
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A(i, j), 15<br />
A ≤ B, 19<br />
A ≤ B, 35<br />
A Λ(i, j), 37<br />
A k , 89<br />
A + k , 90<br />
C(K), 16<br />
C 0(R), 29<br />
C ∞(R), 19, 26, 42<br />
C b (R), 31<br />
C L(Λ), 87<br />
C + k , 90<br />
C l (Λ L), 95<br />
E n(HΛ X L<br />
), 41<br />
F, 14<br />
GE Λ (n, m), 81<br />
G Λ z , 41<br />
H (L) , 53<br />
H 0, 13<br />
H Λ, 36<br />
H Λ(V Λ, V j = a), 46<br />
HΛ X , 41<br />
HΛ D , 39<br />
HΛ N , 39<br />
H Λ(V Λ), 44<br />
H ac, 67<br />
H pp, 67<br />
H sc, 67<br />
H ac, 67<br />
H pp, 67<br />
H sc, 67<br />
L k , 85<br />
l 2 (Z d ), 13<br />
l 2 0(Z d ), 21<br />
N(E), 30<br />
N(HΛ X , E), 42<br />
N Λ(E), 43<br />
n(λ), 43<br />
n Λ(i), 37<br />
P 0, 20<br />
P-almost all, 9<br />
P-almost surely, 9<br />
supp P 0, 20<br />
U i, 25<br />
V ω (L) , 53<br />
V Λ, 44<br />
Γ Λ, 36<br />
Γ D Λ , 38<br />
Γ N Λ , 38<br />
(γ, E)-good, 81<br />
Index<br />
127<br />
γ-good, 82<br />
γ ∞, 97<br />
γ k , 97<br />
δ i, 14<br />
ε(H), 18, 67<br />
ε g(H), 61<br />
Λ L, 13<br />
Λ L(n 0), 13<br />
ν L, 29<br />
ν ac, 66<br />
ν pp, 66<br />
ν sc, 66<br />
˜ν L, 41<br />
˜ν L D , 41<br />
˜ν L N , 41<br />
˜ν L X , 41<br />
ρ(A), 15<br />
σ(A), 15<br />
σ ac(H), 67<br />
σ dis (A), 18<br />
σ ess(A), 18<br />
σ pp(H), 67<br />
σ sc(H), 67<br />
χ Λ, 29<br />
χ M , 17<br />
⋐, 86<br />
∂Λ, 36<br />
∂ Λ2 Λ 1, 39<br />
∂ + Λ, 36<br />
∂ − Λ, 36<br />
∂ − L<br />
Λ, 87<br />
|A|, 13, 81<br />
|Λ L|, 29<br />
|| n|| ∞, 13<br />
|| n|| 1, 13<br />
|| u||, 13<br />
adjacency matrix, 37<br />
alloy-type potential, 9<br />
<strong>An</strong>derson delocalization, 75<br />
<strong>An</strong>derson localization, 75<br />
<strong>An</strong>derson model, 15<br />
Borel-Cantelli lemma, 20<br />
bound state, 68<br />
boundary, 36<br />
bounded below, 18<br />
bounded Borel measure, 65<br />
canonical probability space, 23<br />
characteristic function, 17, 29
128<br />
collection of L−cubes <strong>in</strong>side Λ, 87<br />
cont<strong>in</strong>uous case, 10<br />
coord<strong>in</strong>ation number, 37<br />
count<strong>in</strong>g measure, 8<br />
cyl<strong>in</strong>der sets, 23<br />
density of states, 43<br />
density of states measure, 30<br />
Dirichlet Laplacian, 38<br />
Dirichlet-Neumann bracket<strong>in</strong>g, 35<br />
discrete case, 10<br />
discrete spectrum, 18<br />
dist, 16, 87<br />
distribution, 20<br />
dynamical delocalization, 80<br />
dynamical localization, 78<br />
E-resonant, 94<br />
ergodic, 24<br />
ergodic opera<strong>to</strong>rs, 25<br />
essential spectrum, 18<br />
event, 20<br />
exponential decay, 81<br />
exponential localization, 75<br />
extended states, 75<br />
f<strong>in</strong>itely degenerate, 18<br />
Fourier transform, 14<br />
free opera<strong>to</strong>r, 7<br />
generalized eigenfunction, 61<br />
generalized eigenvalue, 61<br />
geometric resolvent equation, 40<br />
graph Laplacian, 14, 37<br />
Green’s functions, 41<br />
identically distributed, 20<br />
iid, 20<br />
<strong>in</strong>dependent, 20<br />
<strong>in</strong>itial length, 85<br />
<strong>in</strong>ner boundary, 36<br />
<strong>in</strong>tegrated density of states, 30<br />
<strong>in</strong>variant, 24<br />
isolated, 18<br />
kernel, 15<br />
Laplacian, discrete, 14<br />
length scale, 85<br />
Lifshitz behavior, 51<br />
Lifshitz tails, 51<br />
matrix entry, 15<br />
measure<br />
pure po<strong>in</strong>t, 66<br />
absolutely cont<strong>in</strong>uous, 66<br />
bounded Borel, 66<br />
cont<strong>in</strong>uous, 66<br />
positive, 66<br />
s<strong>in</strong>gular cont<strong>in</strong>uous, 66<br />
measure preserv<strong>in</strong>g transformation, 23<br />
m<strong>in</strong>-max pr<strong>in</strong>ciple, 18<br />
mobility edge, 76<br />
multiplicity, 18<br />
multiscale analysis - strong form, 86<br />
multiscale analysis - weak form, 85<br />
Neumann Laplacian, 37<br />
non degenerate, 18<br />
outer boundary, 36<br />
Poisson model, 9<br />
Poisson random measure, 9<br />
polynomially bounded, 61<br />
positive Borel measure, 66<br />
positive opera<strong>to</strong>r, 18, 35<br />
projection valued measure, 17<br />
Radon-Nikodym, 66<br />
RAGE-theorem, 67<br />
random po<strong>in</strong>t measure, 8<br />
random variable, 19<br />
resolvent, 15<br />
resolvent equations, 15<br />
resolvent set, 15<br />
resonant, 94<br />
scatter<strong>in</strong>g states, 70<br />
Schröd<strong>in</strong>ger opera<strong>to</strong>r, 7<br />
simple, 18<br />
simple boundary conditions, 36<br />
s<strong>in</strong>gle site potential, 8<br />
spectral localization, 77<br />
spectral measure<br />
projection valued, 17, 62, 66<br />
real valued, 62<br />
spectral measure zero, 61<br />
spectrum, 15<br />
absolutely cont<strong>in</strong>uous, 67<br />
pure po<strong>in</strong>t , 67<br />
s<strong>in</strong>gular cont<strong>in</strong>uous, 67<br />
s<strong>to</strong>chastic process, 23<br />
S<strong>to</strong>ne-Weierstraß Theorem, 19<br />
support, 20<br />
Temple’s <strong>in</strong>equality, 52<br />
thermodynamic limit, 29<br />
vague convergence, 29
weak convergence, 31<br />
Wegner estimate, 43<br />
well <strong>in</strong>side, 86<br />
Weyl criterion, 22<br />
Weyl sequence, 22<br />
Wiener’s Theorem, 71<br />
129