Tunneling effect in two dimensions with vanishing magnetic fields

TL;DR

This paper investigates the semiclassical behavior of a 2D magnetic Schrödinger operator with a vanishing magnetic field along a symmetric curve, demonstrating tunneling effects and providing explicit eigenvalue splitting formulas.

Contribution

It introduces a new analysis of tunneling in magnetic Schrödinger operators with vanishing fields along symmetric curves, including explicit eigenvalue splitting formulas.

Findings

Proves semi-classical tunneling occurs in the specified setting.

Derives an explicit formula for eigenvalue splitting.

Provides a detailed analysis of the magnetic Schrödinger operator with vanishing magnetic fields.

Abstract

In this paper, we consider the semiclassical 2D magnetic Schr{\"o}dinger operator in the case where the magnetic field vanishes along a smooth closed curve. Assuming that this curve has an axis of symmetry, we prove that semi-classical tunneling occurs. The main result is an expression the splitting of the first two eigenvalues and an explicit tunneling formula.