Poisson wave trace formula for perturbed Dirac operators

TL;DR

This paper develops a Poisson wave trace formula for resonances of perturbed three-dimensional Dirac operators, providing bounds and estimates that connect spectral properties with scattering data.

Contribution

It introduces a global Poisson wave trace formula for Dirac operators with compactly supported perturbations, linking resonances to spectral and scattering characteristics.

Findings

Upper bound on the number of resonances in disks

Estimate on the scattering determinant

Lifshits-Krein trace formula for Dirac operators

Abstract

We consider self-adjoint Dirac operators $\ham{D}=\ham{D}_0 + V(x)$, where $\ham{D}_0$ is the free three-dimensional Dirac operator and $V(x)$ is a smooth compactly supported Hermitian matrix. We define resonances of $\ham{D}$ as poles of the meromorphic continuation of its cut-off resolvent. An upper bound on the number of resonances in disks, an estimate on the scattering determinant and the Lifshits-Krein trace formula then leads to a global Poisson wave trace formula for resonances of $\ham{D}$.