Tunneling effect between radial electric wells in a homogeneous magnetic field

TL;DR

This paper derives an asymptotic tunneling formula for the eigenvalue splitting in a 2D Schrödinger operator with radially symmetric double-well potential under a magnetic field, highlighting quantum tunneling effects.

Contribution

It introduces a new tunneling formula for symmetric double-well potentials with magnetic fields in two dimensions, extending previous results to include magnetic effects.

Findings

Eigenvalue difference is exponentially small in the semiclassical limit.

Derived an explicit asymptotic formula for the eigenvalue splitting.

Extended tunneling analysis to magnetic field scenarios.

Abstract

We establish a tunneling formula for a Schr\"odinger operator with symmetric double-well potential and homogeneous magnetic field, in dimension two. Each well is assumed to be radially symmetric and compactly supported. We obtain an asymptotic formula for the difference between the two first eigenvalues of this operator, that is exponentially small in the semiclassical limit.