  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.
