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Quantum tunneling in deep potential wells and strong magnetic field revisited

Bernard Helffer and Ayman Kachmar

Vol. 6 (2024), No. 2, 319–352
Abstract

We investigate a Hamiltonian with a symmetric double well and a uniform magnetic field, where tunneling occurs in the simultaneous limit of strong magnetic field and deep potential wells with disjoint supports. We derive an accurate estimate of its magnitude, and obtain a precise leading order asymptotic expression for the effect of a strong magnetic field, improving on the upper and lower bounds established earlier by Fefferman, Shapiro and Weinstein.

Keywords
Schrödinger operators, semiclassical techniques, WKB methods, quantum tunneling, magnetic Laplacian
Mathematical Subject Classification
Primary: 35P20, 81Q10, 81U26
Secondary: 82D20
Milestones
Received: 6 January 2023
Revised: 22 November 2023
Accepted: 2 January 2024
Published: 16 May 2024
Authors
Bernard Helffer
Laboratoire de Mathématiques Jean Leray, CNRS
Nantes Université
Nantes
France
Ayman Kachmar
School of Science and Engineering
The Chinese University of Hong Kong (Shenzhen)
Shenzhen
China
Department of Mathematics
Lebanese University
Nabatieh
Lebanon