Low-lying eigenvalues in the semiclassical limit of a Schrödinger operator with an inverse square potential, and non-asymptotic a-zeros of Kummer functions

2025/11/25 by Vanlaere, Roman
#33C15 #34B09 #34L05 #34L15 #34L40 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.2511.20025

Abstract

We provide a precise description of the bottom of the spectrum in the semiclassical limit of a harmonic-type Schrödinger operator with an inverse square potential. By exploiting the connection between the eigenfunctions of these operators and the Kummer and Whittaker functions, we derive accurate localization results for the non-asymptotic zeros of these functions with respect to their first parameter, uniformly with respect to the argument taken large and real. Moreover, our operators are linked to the magnetic Dirichlet Laplacian with a constant magnetic field, so that our results describe its spectrum. Our spectral analysis relies on a WKB-type approach.

Citations

• Flux effects on Magnetic Laplace and Steklov eigenvalues in the exterior of a disk
• On the magnetic Dirichlet to Neumann operator on the exterior of the disk -- diamagnetism, weak-magnetic field limit and flux effects
• Quantum Tunneling and the Aharonov-Bohm effect
• The magnetic Laplacian on the Disc for strong magnetic fields
• Counting eigenvalues below the lowest Landau level
• On the Laplace operator with a weak magnetic field in exterior domains
• Null-controllability properties of the generalized two-dimensional Baouendi-Grushin equation with non-rectangular control sets
• On the semi-classical analysis of the groundstate energy of the Dirichlet Pauli operator
• Polya's conjecture in the presence of a constant magnetic field
• Carleman Estimates for a Class of Degenerate Parabolic Operators

Related

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• Sharp semiclassical spectral asymptotics for local magnetic Schrödinger operators on ℝ^d without full regularity
• Low-energy eigenstates in a vanishing magnetic field
• Semiclassical analysis of Schrödinger operators with magnetic wells
• Semiclassical spectral asymptotics for a magnetic Schrödinger operator with non-vanishing magnetic field