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.

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