Heat traces and existence of scattering resonances for bounded potentials

TL;DR

This paper proves that in odd dimensions, any bounded, compactly supported potential has at least one scattering resonance, extending previous results to less smooth potentials using heat kernel trace asymptotics.

Contribution

It establishes the existence of scattering resonances for bounded potentials in odd dimensions without requiring smoothness, using heat kernel trace asymptotics as a key tool.

Findings

Bounded, compactly supported potentials in odd dimensions have at least one scattering resonance.

The heat kernel trace asymptotic expansion characterizes potential smoothness.

Previous results required smoothness; this work removes that restriction.

Abstract

We show that, in odd dimensions, any real valued, bounded potential of compact support has at least one scattering resonance. For dimensions three and higher this was previously known only for sufficiently smooth potentials. The proof is based on an inverse result, which states that the trace of the associated heat kernel has an appropriate asymptotic expansion if and only if the potential is smooth.