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 λk,V and the corresponding eigenfunctions ψ(⋅;λk,V) of a finite-volume Anderson model (discrete Schrodinger operator) \(\mathcal{H}_{V}= \kappa \Delta_{V}+\xi(\cdot)\) on the multidimensional lattice torus V increasing to the whole of lattice ℤν, provided the distribution function F(⋅) of i.i.d. potential ξ(⋅) satisfies condition −log(1−F(t))=o(t3) and some additional regularity conditions as t→∞. For z∈V, denote by λ0(z) the principal eigenvalue of the “single-peak” Hamiltonian κΔV+ξ(z)δz in l2(V), and let \(\lambda^{0}_{k,V}\) be the kth largest value of the sample λ0(⋅) in V. We first show that the eigenvalues λk,V are asymptotically close to \(\lambda^{0}_{k,V}\). We then prove extremal type limit theorems (i.e., Poisson statistics) for the normalized eigenvalues (λk,V−BV)aV, where the normalizing constants aV>0 and BV are chosen the same as in the corresponding limit theorems for \(\lambda^{0}_{k,V}\). The eigenfunction ψ(⋅;λk,V) is shown to be asymptotically completely localized (as V↑ℤ) at the sites zk,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 Schrodinger operator when potential peaks are sparse.