Localization for Schrödinger Operators with Effective Barriers

Abstract

Our goal is to show that large classes of Schrödinger operatorsH=−Δ+VinL2(Rd) exhibit intervals of dense pure point spectrum, in any dimensiond. We approach this by assuming that the potentialV(x) coincides with a potentialV0(x) of a “comparison operator”H0=−Δ+V0on a sequence of ring shaped (but nog necessarily spherical) regionsUn,n=1,2,… . For energies in the resolvent setρ(H0) ofH0the regionsUnact as “effective barriers” in the sense of quantum mechanical scattering under the potentialV. Under certain assumptions on the geometry of theUnand their complements we show that (i)σac(H(λ))∩ρ(H0)=∅ for everyλ∈R, and (ii)σc(H(λ))∩ρ(H0)=∅ for almost everyλ∈Rwith respect to Lebesgue measure. Hereσacandσcdenote the absolutely continuous and continuous spectrum, respectively, andH(λ) is a “local randomization” ofH, i.e.,H(λ)=H+λW, whereWis any continuous and compactly supported perturbation of fixed sign. Our assumptions leave plenty of room for examples where the spectrum ofHfills entire spectral gaps ofH0. This leads to intervals of dense pure point spectrum forH(λ). We also give an explicit decay estimate for eigenfunctions, thus establishing localization forH(λ) in arbitrary spectral gaps ofH0.