Spectral Theory of Random Schrödinger Operators

The present chapter is devoted to the introduction of the notation, the definitions and most of the results from functional analysis which will be needed in the sequel. Because of lack of space, we refrain from explaining the motivations behind the numerous definitions we introduce. We merely illustrate them with examples of Schrödinger operators and we postpone a more detailed study to Chapter II. Rather than a serious introduction to the spectral theory of self-adjoint operators, this chapter should be understood as a glossary and a summary of the terms and results to be used in the sequel.

This chapter can be viewed as a crash course on Schrödinger operators. We discuss their definitions and some of their spectral properties. We illustrate the abstract theory of Chapter I by the concrete examples of the generalized Laplacian operators in Section II.1 and their perturbations by multiplication operators in Section II.2. The latter are of special importance since they are the Schrödinger operators we want to study. We consider the problem of the essential self-adjointness of these perturbations. We devote a subsection to the properties of the trace class perturbation theory in connection with the stability of the absolutely continuous spectrum. Obviously, the choice of the models we consider and of the techniques we use is strongly biased by the applications we contemplate. We consider only the so-called two-body Hamiltonians and we choose to give a probabilistic proof whenever such a proof is available. It turns out that, once properly defined as self-adjoint operators, Schrödinger operators generate semigroups (when they are bounded below) and unitary groups (when they are defined on the lattice ℤ^d) which can be written using path integrals. We discuss the relevant formulas in Section II.3 of this chapter. They will be very useful in the investigation of the so-called integrated density of states in Chapter VI. Finally, we use these path integral methods to show the regularity properties of the semigroups and the decay of the L²-eigenfunctions.

This chapter is devoted to the investigation of the spectral properties of one-dimensional Schrödinger operators. They deserve a special treatment for the theory of ordinary differential equations ( O.D.E. for short) and its lattice analog, i.e. the theory of finite difference equations, add to the menagerie of tools that can be used. The main new input is the systematic use of the properties of the propagators of the O.D.E. or the transfer matrices in the lattice case. This will be particularly important in the random case which we will study in the following chapters. We also add a special section on quasi one-dimensional systems to discuss some models which are not formally one-dimensional but which can be treated by the same techniques. They usually go under the name of band or strip.

Part of the one dimensional or quasi-one dimensional theory of localization can be reduced to the study of products of random matrices. One of the most important result in this direction is the extension to matrix valued random variables of the strong law of large numbers. Unfortunately the identification of the limit (called the Lyapunov exponent) is more complicated than in the classical case of real valued random variables. In particular this limit can no longer be written as a single expectation. Moreover its determination involves the computation of some invariant measure on the projective space. We only assume that the reader has a minimal background in classical probability theory. Most of the material presented in this chapter is self contained.

Throughout this chapter (Ω,F) is a fixed measurable space and H is a separable complex Hilbert space on which all the operators to be considered are defined. We want to investigate the measurability properties of operator valued functions of the form: from Ω into the space of operators on H We are mainly interested in self-adjoint operators and, as we explained in Chapter I, mostly in are unbounded ones. With this in mind we choose a notion of measurability of operator valued functions which relies on the functional calculus of self-adjoint operators and on the properties of bounded functions of these possibly unbounded operators. We believe that all the technical difficulties relative to these measurability problems have been carefully swept under the rug in most of the research litterature. This is one of the reasons why we decided to study these problems thoroughly.

This chapter is entirely devoted to the study of the so-called integrated density of states. It is an object of interest for various reasons. It is of physical importance for it can be measured experimentally in some cases. On the top of its physical appeal, the integrated density of states is a very interesting mathematical object which deserves to be investigated for its own sake. Many challenging mathematical problems remain open in this respect. Finally, several proofs of technical results crucial to the study of the spectral properties of the random Hamiltonians, and in particular of the localization, rely very heavily on estimates of the integrated density of states.

We assume that a complete dynamical system(Ω,F,ℙ, {θ _t ;t ∈ T} is fixed throughout the chapter. In other words, we assume that (Ω,F,ℙ) is a complete probability space and {θ _t ;t ∈ T} is an ergodic group of automorphisms of the measurable space (Ω,F) which leave invariant the measure ℙ. Except for the short last section, we shall not consider the multidimensional case and we concentrate on the cases T = ℝ and T = ℤ. We will add most of the time the assumption of ergodicity of the flow {θ _t ;t ∈ T} of transformations. All the potential functions considered in this chapter are assumed to be defined on such a dynamical system. They are even assumed to be of the form V(t,ω) = υ(θ _t ω) for some measurable function υ : Ω → ℝ whenever possible.

It was first claimed by P.W. Anderson (1958) in [8] for the multidimensional random Schrödinger operator on a lattice, (associated with i.i.d. potentials) that the spectrum ought to be pure point with exponentially decaying eigenfunctions for a “typical sample” and for “large disorder”. It was later conjectured by Mott & Twose in [249] that this property should hold in the one dimensional case at any disorder. This chapter is devoted to the proof of this last conjecture which we will extend to quasi-one dimensional systems.

Let β be a nonzero real number and let H(ω) be the random Schrödinger operator on ℤ^d defined by where {V(x) ; x ∈ ℤ^d} is an i.i.d. family of potentials defined on some probability space (Ω,ℙ). Such a model proposed by Anderson in [8], is usually refered to as the “Anderson model”. It has been shown in the preceding chapters that the behavior at infinity of the solutions of the “eigenvalue equation” is crucial in the study of spectral properties of the operator H(ω). The behavior at infinity of the Green’s function G(⋋,x,y) which satisfies the above equation for any x ≠ y is also of primordial interest. The links between the exponential growth of the solutions of the eigenvalue equation and the exponential decay of the Green’s function have already been pointed out in the one dimensional case.