Universality Limits of a Reproducing Kernel for a Half-Line Schrödinger Operator and Clock Behavior of Eigenvalues

Abstract

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 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 ξ 0 ∈ I, then uniformly in Iwhere 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 Lρ . We adapt the methods used to show similar results for orthogonal polynomials.vi

Level repulsion for Schrödinger operators with singular continuous spectrum

“…Results for one-dimensional operators with random decaying potentials [6,16,17,27] seem to indicate a pattern whereby greater continuity of the spectral measure corresponds to greater repulsion. Deterministic results establishing asymptotic regular spacing for absolutely continuous measures [2,10,19,20,21,22,23,24,29,30,32] Our aim in the present paper is to show that the situation is more subtle than what may be thought in light of the discussion above. We shall present a family of half-line Schrödinger operators with purely singular continuous spectrum on the positive half-line, whose finitevolume eigenvalues display clock asymptotic behavior (see Definition 1.1 below), which is a very strong form of repulsion.…”

“…Results for one-dimensional operators with random decaying potentials [6,16,17,27] seem to indicate a pattern whereby greater continuity of the spectral measure corresponds to greater repulsion. Deterministic results establishing asymptotic regular spacing for absolutely continuous measures [2,10,19,20,21,22,23,24,29,30,32] work in the same vein.…”

“…The problem of understanding the asymptotics of finite-volume level spacings for Schrödinger operators has been receiving a considerable amount of attention in recent years ( [1,2,3,5,6,10,11,14,16,17,19,20,21,22,23,24,25,26,27,29,30,32,33] is a small subset of relevant references). While results in the multidimensional setting were obtained predominantly for random operators, in the one-dimensional case both deterministic and random operators were studied.…”

“…This object, introduced in [23], is the continuous analog of the classical CD kernel. The CD kernel arises naturally in the study of orthogonal polynomials and has a wide range of applications (see [31] for a survey).…”

An Update on Local Universality Limits for Correlation Functions Generated by Unitary Ensembles

“…Anna Maltsev [65] observed and used analogies between Schrödinger operators and orthogonal polynomials to establish universality limits for Schrödinger operators. Let…”