Double-wells in a strong magnetic field: absence of tunneling for infinitely many values of the coupling constant
Year: 2026
Type: preprint
Abstract: 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 $L\ge L_0$ the associated magnetic double-well Hamiltonian has infinitely many exact degeneracies of its two lowest eigenvalues as $λ\to\infty$. 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.
Source: arXiv (Cornell University)
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Awards: G1190173965, Waves, Novel Two-Dimensional Materials, and Applications, AIMing: Discovery through Machine Learning in Partial Differential Equations, Stabilization of nonlinear partial differential equations: theory and methods, FRG: Collaborative Research: Singularities in Incompressible Flows: Computer Assisted Proofs and Physics-Informed Neural Networks, Hi! PARIS AI cluster 2030 for Science, Business and Society, G4510828401, Fourier analysis and partial differential equations, RTG: Research Training in Applied Mathematics at Columbia University, G7080048889, Waves, Quantum Materials, Metamaterials, and Applications, Classification of Topological Insulators, G829089859
OpenAlex ID: W7213939620