Eigenvalue spacing for 1D singular Schrödinger operators

Article type: Research Article

Authors: Hillairet, Luc^{a; *} | Marzuola, Jeremy L.^b

Affiliations: [a] Institut Denis Poisson, Université d’Orléans – Université de Tours – CNRS, UMR 7013, Orléans, France | [b] Mathematics Department, University of North Carolina, CB#3255, Phillips Hall, Chapel Hill, NC, USA

Correspondence: [*] Corresponding author. E-mail: luc.hillairet@univ-orleans.fr.

Abstract: The aim of this paper is to provide uniform estimates for the eigenvalue spacing of one-dimensional semiclassical Schrödinger operators with singular potentials on the half-line. We introduce a new development of semiclassical measures related to families of Schrödinger operators that provides a means of establishing uniform non-concentration estimates within that class of operators. This dramatically simplifies analysis that would typically require detailed WKB expansions near the turning point, near the singular point and several gluing type results to connect various regions in the domain.

Keywords: Schrödinger operators, spectral asymptotics, semiclassical techniques, uniform first order WKB expansions

DOI: 10.3233/ASY-221814

Journal: Asymptotic Analysis, vol. 133, no. 1-2, pp. 267-289, 2023