Dynamics of Partial Differential Equations

Volume 11 (2014)

Number 4

Existence of Dirac resonances in the semi-classical limit

Pages: 381 – 395

DOI: https://dx.doi.org/10.4310/DPDE.2014.v11.n4.a5

Authors

J. Kungsman (Department of Mathematics, Uppsala University, Uppsala, Sweden)

M. Melgaard (Department of Mathematics, University of Sussex, Brighton, United Kingdom)

Abstract

We study the existence of quantum resonances of the three-dimensional semiclassical Dirac operator perturbed by smooth, bounded and real-valued scalar potentials $V$ decaying like ${\langle x \rangle}^{-\delta}$ at infinity for some $\delta \gt 0$. By studying analytic singularities of a certain distribution related to $V$ and by combining two trace formulas, we prove that the perturbed Dirac operators possess resonances near $\sup V + 1$ and $\inf V - 1$. We also provide a lower bound for the number of resonances near these points expressed in terms of the semiclassical parameter.

Keywords

resonance, Dirac operator, trace formulas

2010 Mathematics Subject Classification

Primary 35P25, 35Q40. Secondary 34C40, 81Q20.

Published 12 December 2014