Double-wells in a strong magnetic field: absence of tunneling for infinitely many values of the coupling constant
Abstract
We prove the fixed-potential conjecture of Fefferman, Shapiro, and Weinstein for inversion-symmetric magnetic double wells. We construct a number L0>0 and one smooth, compactly supported, nonpositive, nonradial single-well potential v, independent of both the coupling parameter λ and the well displacement, such that for every fixed L≥ L0 the associated magnetic double-well Hamiltonian has infinitely many exact degeneracies of its two lowest eigenvalues as λ→∞. The magnetic hopping coefficient also vanishes along an infinite sequence tending to infinity. The potential consists of a radial core and two small, reflection-related perturbations supported on one-sided quadratic cusps. Their log-flat profiles isolate two dominant hopping contributions, which we evaluate by steepest descent after inserting the radial ground-state asymptotics. The resulting asymptotic cosine expression, together with error bounds for the even--odd splitting, yields infinitely many exact eigenvalue crossings and changes of ground-state parity.
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BibTeX
@misc{buck2026doublewells,
title = {{Double-wells in a strong magnetic field: absence of tunneling for infinitely many values of the coupling constant}},
author = {Kevin Buck and Charles L. Fefferman and Javier Gómez-Serrano and Amaury Hayat and Jacob Shapiro and Michael I. Weinstein},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.21078},
eprint = {2609.21078},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.21078},
}
RIS
TY - UNPB
TI - Double-wells in a strong magnetic field: absence of tunneling for infinitely many values of the coupling constant
AU - Kevin Buck
AU - Charles L. Fefferman
AU - Javier Gómez-Serrano
AU - Amaury Hayat
AU - Jacob Shapiro
AU - Michael I. Weinstein
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2609.21078
UR - https://arxiv.org/abs/2609.21078
AB - We prove the fixed-potential conjecture of Fefferman, Shapiro, and Weinstein for inversion-symmetric magnetic double wells. We construct a number L_0>0 and one smooth, compactly supported, nonpositive, nonradial single-well potential v, independent of both the coupling parameter λ and the well displacement, such that for every fixed L≥ L_0 the associated magnetic double-well Hamiltonian has infinitely many exact degeneracies of its two lowest eigenvalues as λ→∞. The magnetic hopping coefficient also vanishes along an infinite sequence tending to infinity. The potential consists of a radial core and two small, reflection-related perturbations supported on one-sided quadratic cusps. Their log-flat profiles isolate two dominant hopping contributions, which we evaluate by steepest descent after inserting the radial ground-state asymptotics. The resulting asymptotic cosine expression, together with error bounds for the even--odd splitting, yields infinitely many exact eigenvalue crossings and changes of ground-state parity.
ER -