Flux and symmetry effects on quantum tunneling

TL;DR

This paper develops an abstract spectral reduction framework for self-adjoint operators, illustrating flux and symmetry effects on quantum tunneling through three applications involving magnetic fields and potential wells.

Contribution

Introduces a novel spectral reduction framework and applies it to analyze flux effects and tunneling phenomena in magnetic and potential well systems.

Findings

Flux effects cause eigenvalue crossings in the studied systems.

Extended the validity of tunneling approximations in magnetic double well potentials.

Ruling out artificial conditions improves understanding of electromagnetic tunneling.

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.