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• W2108727307 abstract "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 show that if solutions u(ξ, x) are bounded in x by $${e^{epsilon x}}$$ 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 $${xi_0in I}$$ , then uniformly in I, $$frac{S_L(xi_0 + a/L, xi_0 + b/L)}{S_L(xi_0, xi_0)}rightarrow frac{sin(pirho(xi_0)(a - b))}{pirho(xi_0)(a - b)},$$ where 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 $${frac{1}{Lrho}}$$ . We adapt the methods used to show similar results for orthogonal polynomials." @default.
• W2108727307 created "2016-06-24" @default.
• W2108727307 creator A5020802578 @default.
• W2108727307 date "2010-06-26" @default.
• W2108727307 modified "2023-09-27" @default.
• W2108727307 title "Universality Limits of a Reproducing Kernel for a Half-Line Schrödinger Operator and Clock Behavior of Eigenvalues" @default.
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• W2108727307 doi "https://doi.org/10.1007/s00220-010-1074-z" @default.
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