Universality Limits of a Reproducing Kernel for a Half-Line Schrödinger 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 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 showthat 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 ∈ I, then uniformly in I, SL(ξ_0 + a/L, ξ_0 + b/L)/S_L (ξ_0, ξ_0) → 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 1/L_ρ. We adapt the methods used to show similar results for orthogonal polynomials.