Universality limits of a reproducing kernel for a half-line Schrodinger operator and clock behavior of eigenvalues

Anna V Maltsev

Research output: Contribution to journalArticle (Academic Journal)peer-review

3 Citations (Scopus)

Abstract

We extend some recent results of Lubinsky, Levin, Simon, and Totik from measures with compact support to spectral measures of Schrödinger operators on the half-line. In particular, we define a reproducing kernel SL for Schrödinger operators and we use it to study the fine spacing of eigenvalues in a box of the half-line Schrödinger operator with perturbed periodic potential. We show that if solutions u(ξ, x) are bounded in x by uniformly for ξ near the spectrum in an average sense and the spectral measure is positive and absolutely continuous in a bounded interval I in the interior of the spectrum with TeX, then uniformly in I,TeXwhere ρ(ξ)dξ is the density of states. We deduce that the eigenvalues near ξ0 in a large box of size L are spaced asymptotically as TeX. We adapt the methods used to show similar results for orthogonal polynomials.
Original languageEnglish
Pages (from-to)461-484
Number of pages24
JournalCommunications in Mathematical Physics
Volume298
Issue number2
DOIs
Publication statusPublished - 1 Sept 2010

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