An Invitation to Random SchrÂ¨odinger operators - FernUniversitÃ¤t in ...

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12 Notes and Remarks Standard references for mathematical methods of quantum mechanics are [57], [117], [114], [116], [115] and [141], [30]. Most of the necessary prerequisites from spectral theory can be found in [117] or [141]. A good source for the probabilistic background is [95] and [96]. The physical theory of random Schrödinger operators is described in [9], [99], [135] and [136]. References for the mathematical approach to random Schrödinger operators are [23], [30], [97], [58], [112] and [128].
13 3. Setup: The Anderson model 3.1. Discrete Schrödinger operators. In the Anderson model the Hilbert space L 2 (R d ) is replaced by the sequence space l 2 (Z d ) = {(u i ) i∈Z d| ∑ |u i | 2 < ∞} (3.1) i∈Z d = {u : Z d → C ∣ ∣ ∑ n∈Z d |u(n)| 2 < ∞} . (3.2) We denote the norm on l 2 (Z d ) by || u|| = ( ∑ n∈Z d |u(n)| 2 ) 1 2 (3.3) Here, we think of a particle moving on the lattice Z d , so that in the case ‖u‖ = 1 the probability to find the particle at the point n ∈ Z d is given by |u(n)| 2 . Note, that we may think of u either as a function u(n) on Z d or as a sequence u n indexed by Z d . It will be convenient to equip Z d with two different norms. The first one is || n || ∞ := sup | n ν | . (3.4) ν=1,...,d This norm respects the cubic structure of the lattice Z d . For example, it is convenient to define the cubes (n 0 ∈ Z d , L ∈ N) Λ L (n 0 ) := {n ∈ Z d ; || n − n 0 || ∞ ≤ L} . (3.5) Λ L (n 0 ) is the cube of side length 2L + 1 centered at n 0 . It contains |Λ L (n 0 ) | := (2L + 1) d points. Sometimes we call |Λ L (n 0 ) | the volume of Λ L (n 0 ). In general, we denote by |A| the number of elements of the set A. To shorten notation we write Λ L for Λ L (0). The other norm we use on Z d is || n|| 1 := d∑ | n ν | . (3.6) ν=1 This norm reflects the graph structure of Z d . Two vertices n and m of the graph Z d are connected by an edge, if they are nearest neighbors, i. e. if || n − m|| 1 = 1. For arbitrary n, m ∈ Z d the norm || n − m|| 1 gives the length of the shortest path between n and m. The kinetic energy operator H 0 is a discrete analogue of the (negative) Laplacian, namely
Page 1: An Invitation to Random Schrödinge
Page 4 and 5: 4 10.2. Analytic estimate - first t
Page 7 and 8: 7 2. Introduction: Why random Schr
Page 9 and 10: is the random point measure ν =
Page 11: symmetric subspace is the correct o
Page 15 and 16: orthonormal basis of l 2 (Z d ). Th
Page 17 and 18: 17 (αf + βg) (A) = αf(A) + βg(A
Page 19 and 20: THEOREM 3.1 (Min-max principle). If
Page 21 and 22: 21 sup |V ω (j)| ≤ M . j∈Z d E
Page 23 and 24: 23 4.1. Ergodic stochastic processe
Page 25 and 26: For a proof of this fundamental res
Page 27: holds for all . Thus the measures
Page 30 and 31: 30 REMARK 5.3. The right hand side
Page 32 and 33: 32 (2) In the continuous case the d
Page 34 and 35: 34 PROOF: If λ /∈ Σ then there
Page 36 and 37: 36 The easiest version of boundary
Page 38 and 39: 38 DEFINITION 5.18. The Dirichlet L
Page 40 and 41: 40 5.3. The geometric resolvent equ
Page 42 and 43: 42 We define the eigenvalue countin
Page 44 and 45: 44 PROOF (Corollary 5.25) : By the
Page 46 and 47: 46 Hence ( ∂ E tr ( ϱ (H Λ (V
Page 48 and 49: 48 We start the proof with the foll
Page 51 and 52: 51 6.1. Statement of the Result. 6.
Page 53 and 54: 53 In fact (H 0 ) N Λ L ψ 0 = 0.
Page 55 and 56: 55 This estimate is the desired upp
Page 57 and 58: for any ψ with ||ψ|| = 1. Now we
Page 59: For an approach using periodic appr
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62 µ n,m (∆) = 〈δ n , µ(∆)
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64 So far, the sequence α n has on
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66 mean in what follows a complex-v
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68 Let us look at the time evolutio
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70 lim t→∞ ( ∑ x∈Λ | e −
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72 → = ∫ T 1 |ˆµ(t)| 2 dt T 0
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75 8.1. What physicists know. 8. An
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of a Schrödinger operator on R d c
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PROOF: From Theorem 7.14 we know
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81 9. The Green’s function and th
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83 |ψ(n)| ≤ ≤ ∣ ∑ (m ′ ,
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where (9.14) holds. This effect is
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87 by C L (Λ) = {Λ L (n) | Λ L (
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89 with some k ′ ≥ k. We conclu
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91 We will prove LEMMA 9.16. If Res
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93 10. Multiscale analysis 10.1. St
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95 If an overwhelming majority of t
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THEOREM 10.5. If the cube Λ L is n
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Thus, (10.28) and (10.30) inserted
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101 |G Λ L E (u, n)| ≤ ∑ (q,q
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(2) no cube Λ 2l (m) in Λ L is E-
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105 ∑ i,j∈Λ L Λ l (i)∩Λ l
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connection with a Wegner-type bound
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109 If n j belongs to one of the M
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≤ ∑ C i ∈ C l (Λ L ) pairwis
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114 F u(n) = F n0 u(n) = e µ || n
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116 11.3. Energies near the bottom
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119 12. Appendix: Lost in Multiscal
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121 References [1] M. Aizenman: Loc
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[51] F. Germinet, A. Klein, J. Sche
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[99] I. M. Lifshits, S. A. Gredesku
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A(i, j), 15 A ≤ B, 19 A ≤ B, 35
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weak convergence, 31 Wegner estimat
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