Exponential Localization for Multi-Dimensional Schrödinger Operator with Random Point Potential

Abstract

We consider a Schrödinger operator -Δα(ω) on L2(ℝd)(d=2,3) whose potential is a sum of point potentials, centered at sites of ℤd, with independent and identically distributed random amplitudes. We prove the existence of the pure point spectrum and the exponential decay of the corresponding eigenfunctions at the negative semi-axis for certain regimes of the disorder. In order to prove localization results, we elucidate the structure of the generalized eigenfunctions of -Δα(ω) and the relation between its negative spectrum and the spectra of a family of infinite-order operators on ℓ2(ℤd). We apply the multiscale analysis scheme to investigate the point spectrum of these operators. We also prove the absolute continuity of the integrated density of states of the operator on the negative part of its spectrum.