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We extend some recent results of Lubinsky, Levin, Simon, and Totik frommeasures with compact support to spectral measures of Schr\"odinger operatorson the half-line. In particular, we define a reproducing kernel $S_L$ forSchr\"odinger operators and we use it to study the fine spacing of eigenvaluesin a box of the half-line Schr\"odinger operator with perturbed periodicpotential. 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 andthe spectral measure is positive and absolutely continuous in a boundedinterval $I$ in the interior of the spectrum with $\xi_0\in I$, then uniformlyin $I$ $$\frac{S_L(\xi_0 + a/L, \xi_0 + b/L)}{S_L(\xi_0, \xi_0)} \to\frac{\sin(\pi\rho(\xi_0)(a - b))}{\pi\rho(\xi_0)(a - b)},$$ 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 fororthogonal polynomials.

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