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


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.


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.))
Subject Keywords:universality limits, spectral theory, Schrodinger operators, eigenvalues in a box
Degree Grantor:California Institute of Technology
Division:Physics, Mathematics and Astronomy
Major Option:Mathematics
Thesis Availability:Public (worldwide access)
Research Advisor(s):
  • Simon, Barry M.
Thesis Committee:
  • Borodin, Alexei
  • Rains, Eric M.
  • Ryckman, Eric
  • Simon, Barry M. (chair)
Defense Date:10 May 2010
Non-Caltech Author Email:annavmaltsev (AT)
Record Number:CaltechTHESIS:05262010-023753573
Persistent URL:
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Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:5840
Deposited By: Anna Maltsev
Deposited On:04 Jun 2010 18:05
Last Modified:22 Aug 2016 21:19

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PDF (universality limits for Schrodinger operators) - Final Version
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