Universality Limits of a Reproducing Kernel for a Half-Line Schr\"odinger Operator and Clock Behavior of Eigenvalues

We extend some recent results of Lubinsky, Levin, Simon, and Totik from measures with compact support to spectral measures of Schr\"odinger operators on the half-line. In particular, we define a reproducing kernel $S_L$ for Schr\"odinger operators and we use it to study the fine spacing of eigenvalues in a box of the half-line Schr\"odinger 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 $I$ $$\frac{S_L(\xi_0 + a/L, \xi_0 + b/L)}{S_L(\xi_0, \xi_0)} \to \frac{\sin(\pi\rho(\xi_0)(a - b))}{\pi\rho(\xi_0)(a - b)},$$ 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.