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

## Citation

Maltsev, Anna (2010) Universality limits of a reproducing kernel for a half-line Schrödinger operator and clock behavior of eigenvalues. Dissertation (Ph.D.), California Institute of Technology. http://resolver.caltech.edu/CaltechTHESIS:05262010-023753573

## 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 show that if solutions $u(\xi, x)$ are bounded in $x$ by $e^{\epsilon x}$ uniformly for $\xi$ 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 $\xi_0\in I$, then uniformly in $I$ $$\frac{S_L(\xi_0 + a/L, \xi_0 + b/L)}{S_L(\xi_0, \xi_0)} \rightarrow \frac{\sin(\pi\rho(\xi_0)(a - b))}{\pi\rho(\xi_0)(a - b)},$$ where $\rho(\xi)d\xi$ is the density of states. We deduce that the eigenvalues near $\xi_0$ in a large box of size $L$ are spaced asymptotically as $\frac{1}{L\rho}$. We adapt the methods used to show similar results for orthogonal polynomials.

Item Type: Thesis (Dissertation (Ph.D.)) universality limits, spectral theory, Schrodinger operators, eigenvalues in a box California Institute of Technology Physics, Mathematics and Astronomy Mathematics Public (worldwide access) Simon, Barry M. Borodin, AlexeiRains, Eric M.Ryckman, EricSimon, Barry M. (chair) 10 May 2010 annavmaltsev (AT) gmail.com CaltechTHESIS:05262010-023753573 http://resolver.caltech.edu/CaltechTHESIS:05262010-023753573 Author No commercial reproduction, distribution, display or performance rights in this work are provided. 5840 CaltechTHESIS Anna Maltsev 04 Jun 2010 18:05 26 Dec 2012 03:26

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