An Invitation to Random Schr¨odinger operators - FernUniversität in ...

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This and many other results can be proven for various types of random Schrödinger
operators. In this lecture we will restrict ourselves to a relatively simple system
known as the Anderson model. Here the Hilbert space is the sequence space
l 2 (Z d ) instead of L 2 (R d ) and the free operator H 0 is a finite-difference operator
rather than the Laplacian. We will call this setting the discrete case in contrast
to Schrödinger operators on L 2 (R d ) which we refer to as the continuous case . In
the references the reader may find papers which extend results we prove here to the
continuous setting.
2.3. The one body approximation.
In the above setting we have implicitly assumed that we describe a single particle
moving in a static exterior potential. This is at best a caricature of what we find in
nature. First of all there are many electrons moving in a solid and they interact with
each other. The exterior potential originates in nuclei or ions which are themselves
influenced both by the other nuclei and by the electrons. In the above discussion
we have also implicitly assumed that the solid we consider extends to infinity in all
directions, (i.e. fills the universe). Consequently, we ought to consider infinitely
many interacting particles. It is obvious that such a task is out of range of the
methods available today. As a first approximation it seems quite reasonable to
separate the motion of the nuclei from the system and to take the nuclei into account
only via an exterior potential. Indeed, the masses of the nuclei are much larger than
those of the electrons.
The second approximation is to neglect the electron-electron interaction. It is not at
all clear that this approximation gives a qualitatively correct picture. In fact, there
is physical evidence that the interaction between the electrons is fundamental for a
number of phenomena.
Interacting particle systems in condensed matter are an object of intensive research
in theoretical physics. In mathematics, however, this field of research is still in its
infancy despite of an increasing interest in the subject.
If we neglect the interactions between the electrons we are left with a system of
noninteracting electrons in an exterior potential. It is not hard to see that such
a system (and the corresponding Hamiltonian) separates, i.e. the eigenvalues are
just sums of the one-body eigenvalues and the eigenfunctions have product form.
So, if ψ 1 , ψ 2 , . . . , ψ N are eigenfunctions of the one-body system corresponding to
eigenvalues E 1 , E 2 , . . . , E N respectively, then
Ψ(x 1 , x 2 , . . . , x N ) = ψ 1 (x 1 ) · ψ 2 (x 2 ) · . . . · ψ N (x N ) . (2.10)
is an eigenfunction of the full system with eigenvalue E 1 + E 2 + . . . + E N .
However, there is a subtlety to obey here, which is typical to many particle Quantum
Mechanics. The electrons in the solid are indistinguishable, since we are unable
to ‘follow their trajectories’. The corresponding Hamiltonian is invariant under
permutation of the particles. As a consequence, the N-particle Hilbert space
consists either of totally symmetric or of totally antisymmetric functions of the particle
positions x 1 , x 2 , . . . , x N . It turns out that for particles with integer spin the