Extremal Theory for Spectrum of Random Discrete Schrödinger Operator. II. Distributions with Heavy Tails

Abstract

We study the asymptotic structure of the first K largest eigenvalues λ _k,V and the corresponding eigenfunctions ψ(⋅;λ _k,V ) of a finite-volume Anderson model (discrete Schrödinger operator) $\mathcal{H}_{V}= \kappa \Delta_{V}+\xi(\cdot)$ on the multidimensional lattice torus V increasing to the whole of lattice ℤ^ν, provided the distribution function F(⋅) of i.i.d. potential ξ(⋅) satisfies condition −log(1−F(t))=o(t ³) and some additional regularity conditions as t→∞. For z∈V, denote by λ ⁰(z) the principal eigenvalue of the “single-peak” Hamiltonian κΔ _V +ξ(z)δ _z in l ²(V), and let $\lambda^{0}_{k,V}$ be the kth largest value of the sample λ ⁰(⋅) in V. We first show that the eigenvalues λ _k,V are asymptotically close to $\lambda^{0}_{k,V}$. We then prove extremal type limit theorems (i.e., Poisson statistics) for the normalized eigenvalues (λ _k,V −B _V )a _V , where the normalizing constants a _V >0 and B _V are chosen the same as in the corresponding limit theorems for $\lambda^{0}_{k,V}$. The eigenfunction ψ(⋅;λ _k,V ) is shown to be asymptotically completely localized (as V↑ℤ) at the sites z _k,V ∈V defined by $\lambda^{0}(z_{k,V})=\lambda^{0}_{k,V}$. Proofs are based on the finite-rank (in particular, rank one) perturbation arguments for discrete Schrödinger operator when potential peaks are sparse.