Citation
Maltsev, Anna Victoria (2010) Universality Limits of a Reproducing Kernel for a HalfLine Schrödinger Operator and Clock Behavior of Eigenvalues. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/QQGJ1A69. https://resolver.caltech.edu/CaltechTHESIS:05262010023753573
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 halfline. 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 halfline 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): 
 
Thesis Committee: 
 
Defense Date:  10 May 2010  
NonCaltech Author Email:  annavmaltsev (AT) gmail.com  
Record Number:  CaltechTHESIS:05262010023753573  
Persistent URL:  https://resolver.caltech.edu/CaltechTHESIS:05262010023753573  
DOI:  10.7907/QQGJ1A69  
ORCID: 
 
Default Usage Policy:  No commercial reproduction, distribution, display or performance rights in this work are provided.  
ID Code:  5840  
Collection:  CaltechTHESIS  
Deposited By:  Anna Maltsev  
Deposited On:  04 Jun 2010 18:05  
Last Modified:  08 Nov 2019 18:10 
Thesis Files

PDF (universality limits for Schrodinger operators)
 Final Version
See Usage Policy. 328kB 
Repository Staff Only: item control page