The wave trace and resonances of the magnetic Hamiltonian with singular vector potentials

TL;DR

This paper analyzes the wave trace singularities of the magnetic Aharonov--Bohm Hamiltonian with multiple solenoids and derives a lower bound on scattering resonances near the real axis.

Contribution

It provides a formula for the wave trace singularities of the Aharonov--Bohm Hamiltonian with multiple solenoids and applies it to estimate resonances.

Findings

Derived a formula for wave trace singularities with multiple solenoids.

Established a lower bound for scattering resonances near the real axis.

Assumed generic non-collinearity of three solenoids for analysis.

Abstract

We study leading order singularities of the wave trace of the Aharonov--Bohm Hamiltonian on $\mathbf{R}^2$ with multiple solenoids under a generic assumption that no three solenoids are collinear. Then we apply our formula to get a lower bound of scattering resonances in a logarithmic neighborhood near the real axis.