Localization for Schrödinger operators with effective barriers

Academic Article

Abstract

  • Our goal is to show that large classes of Schrödinger operators H= - Δ + V in L2(Rd) exhibit intervals of dense pure point spectrum, in any dimension d. We approach this by assuming that the potential V(x) coincides with a potential V0(x) of a "comparison operator" H0 = - Δ + V0 on a sequence of ring shaped (but nog necessarily spherical) regions Un, n = 1, 2, .... For energies in the resolvent set p(H0) of H0 the regions Un act as "effective barriers" in the sense of quantum mechanical scattering under the potential V. Under certain assumptions on the geometry of the Un and their complements we show that (i) σac(H(λ))∩ρ(H0) = ∅ for every λ∈R, and (ii) σc(H(λ))∩ρ(H0) = ∅ for almost every λ∈R with respect to Lebesgue measure. Here σac and σc denote the absolutely continuous and continuous spectrum, respectively, and H(λ) is a "local randomization" of H, i.e., H(λ)= H + λW, where W is any continuous and compactly supported perturbation of fixed sign. Our assumptions leave plenty of room for examples where the spectrum of H fills entire spectral gaps of H0. This leads to intervals of dense pure point spectrum for H(λ). We also give an explicit decay estimate for eigenfunctions, thus establishing localization for H(λ) in arbitrary spectral gaps of H0. © 1997 Academic Press.
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    Published In

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    Author List

  • Stolz G
  • Start Page

  • 416
  • End Page

  • 429
  • Volume

  • 146
  • Issue

  • 2