Limit theorems for basic states of the Anderson model
Let \(V=\{x=(x^ 1,\dots,x^ \nu) \in \mathbb{Z}^ \nu:| x^ i | \leq n\), \(i=1, \dots, \nu\}\), and let \(N=(2n+1)^ \nu\) be the number of points in \(V\), \(\nu \geq 1\). Let us consider the following Hamiltonian in \(L^ 2(V)\): \[ H_ V=\kappa \Delta_ V+\xi (x),\quad x \in V, \tag{1} \] where \(\Delta_ V\) is the discrete Laplacian on \(V\) with zero Dirichlet boundary conditions (the restriction of the operator \(\Delta \varphi (x)=\sum_{| y-x |=1} \varphi (y)\), \(x \in \mathbb{Z}^ \nu\), to \(V)\), \(| x |=| x^ 1 |+\cdots+| x^ \nu |\), \(\xi (x)\), \(x \in \mathbb{Z}^ \nu\), is a random field of independent identically distributed variables, \(\kappa\) is a positive constant. Let \(\lambda_{1,N}>\cdots>\lambda_{N,N}\) be (random) eigenvalues of the operator (1). This paper presents limit theorems for the first (normalized) eigenvalues \(\lambda_{1,N},\dots,\lambda_{i,N}\) under the following assumption on the distribution density \(p(t):=(d/dt) P(\xi (0)<t)\): \[ p(t)=\bigl( A \alpha+o(1) \bigr) t^{\alpha-\beta-1} \exp (- At^ \alpha) \quad (t \to \infty), \tag{2} \] where \(\alpha>0\), \(A>0\), \(- \infty<\beta<\infty\). In the theory of Lifshits tails, this class of distributions is of special importance due to existence of bifurcations with respect to \(\alpha\) in the Anderson model (1).
scientific article; zbMATH DE number 561007