The Aharonov-Bohm effect in spectral asymptotics of the magnetic Schr\"odinger operator

TL;DR

This paper demonstrates that the magnetic potential influences the spectral asymptotics of the Schrödinger operator in an annulus and on a torus, revealing the Aharonov-Bohm effect even without a magnetic field.

Contribution

It provides a spectral analysis showing the Aharonov-Bohm effect manifests through spectral asymptotics in specific geometries without magnetic fields.

Findings

Spectrum depends on the cosine of magnetic flux

Spectral asymptotics are affected by magnetic potential

Aharonov-Bohm effect observed in spectral data

Abstract

We show that in the absence of a magnetic field the spectrum of the magnetic Schr\"odinger operator in an annulus depends on the cosine of the flux associated with the magnetic potential. This result follows from an analysis of a singularity in the wave trace for this Schr\"odinger operator, and hence shows that even in the absence of a magnetic field the magnetic potential can change the asymptotics of the Schr\"odinger spectrum, i.e. the Aharonov-Bohm effect takes place. We also study the Aharonov-Bohm effect for the magnetic Schr\"odinger operator on a torus.