Universality Limits of a Reproducing Kernel for a Half-Line Schrödinger Operator and Clock Behavior of Eigenvalues
- Creators
- Maltsev, Anna
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 S_L 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 showthat if solutions u(ξ,x) are bounded in x by e^(εx) 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 ξ_0 ∈ I, then uniformly in I, SL(ξ_0 + a/L, ξ_0 + b/L)/S_L (ξ_0, ξ_0) → sin(πρ(ξ_0)(a − b))/πρ(ξ_0)(a − b), where ρ(ξ)dξ is the density of states. We deduce that the eigenvalues near ξ_0 in a large box of size L are spaced asymptotically as 1/L_ρ. We adapt the methods used to show similar results for orthogonal polynomials.
Additional Information
© 2010 Springer-Verlag. Received: 31 July 2009. Accepted: 16 March 2010. Published online: 26 June 2010. Communicated by H.-T. Yau. I would like to thank my advisor Professor Barry Simon for all his help. I would also like to thank Professors Jonathan Breuer and Fritz Gesztesy for useful discussions.Additional details
- Eprint ID
- 19384
- Resolver ID
- CaltechAUTHORS:20100811-082529392
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2010-08-13Created from EPrint's datestamp field
- Updated
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2021-11-08Created from EPrint's last_modified field