Flux and symmetry effects on quantum tunneling

Bernard Helffer; Ayman Kachmar; Mikael Persson Sundqvist

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arXiv · 2307.06712

Flux and symmetry effects on quantum tunneling

Bernard Helffer·Ayman Kachmar·Mikael Persson Sundqvist

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Abstract

Motivated by the analysis of the tunneling effect for the magnetic Laplacian, we introduce an abstract framework for the spectral reduction of a self-adjoint operator to a hermitian matrix. We illustrate this framework by three applications, firstly the electro-magnetic Laplacian with constant magnetic field and three equidistant potential wells, secondly a pure constant magnetic field and Neumann boundary condition in a smoothed triangle, and thirdly a magnetic step where the discontinuity line is a smoothed triangle. Flux effects are visible in the three aforementioned settings through the occurrence of eigenvalue crossings. Moreover, in the electro-magnetic Laplacian setting with double well radial potential, we rule out an artificial condition on the distance of the wells and extend the range of validity for a recently established tunneling approximation, thereby settling the problem of electro-magnetic tunneling under constant magnetic field and a sum of translated radial electric potentials.

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BibTeX RIS

Bernard Helffer, Ayman Kachmar, Mikael Persson Sundqvist. 2023-07-13. Flux and symmetry effects on quantum tunneling. https://arxiv.org/abs/2307.06712

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