Spectral Asymptotics of a Singular Harmonic Oscillator with Aharonov--Bohm Flux
Abstract
Motivated by a recent article of R. Vanlaere (2026), we investigate the spectral asymptotics of a singular harmonic oscillator arising from the radial reduction of the magnetic Schrödinger operator on the unit disk in the presence of a constant magnetic field and an Aharonov-Bohm flux. The corresponding eigenvalue problem is equivalent to the study of the zeros of the Kummer confluent hypergeometric function with respect to its first parameter. Combining now standard semiclassical methods with the uniform asymptotic theory of Whittaker and confluent hypergeometric functions developed by Dunster (1989), Gabutti-Gatteschi (2001) and improved quite recently by Dunster (2026), we obtain a comprehensive description of the different spectral regimes. When the boundary lies in the classically forbidden region, we derive the exponentially small asymptotic correction to the eigenvalue caused by the tunneling effect. For energy levels above the tunneling regime, we derive a shifted Bohr-Sommerfeld asymptotics for highly excited states at a fixed magnetic field. Finally, for the transition regime where the energy level touches the boundary, we derive a three-term asymptotic expansion of the eigenvalue. In an appendix, we translate these spectral results into explicit asymptotic formulas for the zeros of the Kummer function.
Related
- Rough spectral asymptotics for commutators of general singular integrals
- Sharp harmonic-mean inequalities for Neumann and Aharonov--Bohm spectra
- Boundary-layer asymptotics for Gaussian-smoothed singular measures
- An unified approach for the (inverted) quantum harmonic oscillator and the Berry-Keating hamiltonian
- Combinatorics of noncommutative harmonic oscillator determinant
BibTeX
@misc{helffer2026spectral,
title = {{Spectral Asymptotics of a Singular Harmonic Oscillator with Aharonov--Bohm Flux}},
author = {Bernard Helffer and Ayman Kachmar and François Nicoleau},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.28076},
eprint = {2609.28076},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.28076},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Spectral Asymptotics of a Singular Harmonic Oscillator with Aharonov--Bohm Flux
AU - Bernard Helffer
AU - Ayman Kachmar
AU - François Nicoleau
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2609.28076
UR - https://arxiv.org/abs/2609.28076
AB - Motivated by a recent article of R. Vanlaere (2026), we investigate the spectral asymptotics of a singular harmonic oscillator arising from the radial reduction of the magnetic Schrödinger operator on the unit disk in the presence of a constant magnetic field and an Aharonov-Bohm flux. The corresponding eigenvalue problem is equivalent to the study of the zeros of the Kummer confluent hypergeometric function with respect to its first parameter. Combining now standard semiclassical methods with the uniform asymptotic theory of Whittaker and confluent hypergeometric functions developed by Dunster (1989), Gabutti-Gatteschi (2001) and improved quite recently by Dunster (2026), we obtain a comprehensive description of the different spectral regimes. When the boundary lies in the classically forbidden region, we derive the exponentially small asymptotic correction to the eigenvalue caused by the tunneling effect. For energy levels above the tunneling regime, we derive a shifted Bohr-Sommerfeld asymptotics for highly excited states at a fixed magnetic field. Finally, for the transition regime where the energy level touches the boundary, we derive a three-term asymptotic expansion of the eigenvalue. In an appendix, we translate these spectral results into explicit asymptotic formulas for the zeros of the Kummer function.
ER -