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arXiv · 0908.1440

Universality Limits of a Reproducing Kernel for a Half-Line Schrödinger Operator and Clock Behavior of Eigenvalues

Anna Maltsev
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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 $u(ξ, x)$ 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 $ξ_0\in I$, then uniformly in $I$ $$\frac{S_L(ξ_0 + a/L, ξ_0 + b/L)}{S_L(ξ_0, ξ_0)} \to \frac{\sin(πρ(ξ_0)(a - b))}{πρ(ξ_0)(a - b)},$$ 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 $\frac{1}{Lρ}$. We adapt the methods used to show similar results for orthogonal polynomials.

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Anna Maltsev. 2009-08-11. Universality Limits of a Reproducing Kernel for a Half-Line Schrödinger Operator and Clock Behavior of Eigenvalues. https://arxiv.org/abs/0908.1440

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