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 SL 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 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 TeX, then uniformly in I,TeXwhere ρ(ξ)dξ is the density of states. We deduce that the eigenvalues near ξ0 in a large box of size L are spaced asymptotically as TeX. We adapt the methods used to show similar results for orthogonal polynomials.