Asian Journal of Mathematics

Volume 25 (2021)

Number 2

Poisson wave trace formula for Dirac resonances at spectrum edges and applications

Pages: 243 – 276

DOI: https://dx.doi.org/10.4310/AJM.2021.v25.n2.a5

Authors

B. Cheng (Department of Mathematics, School of Mathematical and Physical Sciences, University of Sussex, Brighton, United Kingdom)

M. Melgaard (Department of Mathematics, School of Mathematical and Physical Sciences, University of Sussex, Brighton, United Kingdom)

Abstract

We study the self-adjoint Dirac operators $\mathbb{D} = \mathbb{D}_0 + V(x)$, where $\mathbb{D}_0$ is the free three-dimensional Dirac operator and $V(x)$ is a smooth compactly supported Hermitian matrix potential. We define resonances of $\mathbb{D}$ as poles of the meromorphic continuation of its cut-off resolvent. By analyzing the resolvent behaviour at the spectrum edges $\pm m$, we establish a generalized Birman–Krein formula, taking into account possible resonances at $\pm m$. As an application of the new Birman–Krein formula we establish the Poisson wave trace formula in its full generality. The Poisson wave trace formula links the resonances with the trace of the difference of the wave groups. The Poisson wave trace formula, in conjunction with asymptotics of the scattering phase, allows us to prove that, under certain natural assumptions on $V$, the perturbed Dirac operator has infinitely many resonances; a result similar in nature to Melrose’s classic 1995 result for Schrödinger operators.

Keywords

Dirac operators, Birman–Krein formula, Poisson wave trace formula, threshold resolvent behaviour

2010 Mathematics Subject Classification

Primary 35Q41. Secondary 35P25, 47A40, 81U20.

Received 29 October 2019

Accepted 31 July 2020

Published 15 October 2021