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# Dynamics of Partial Differential Equations

## Volume 11 (2014)

### Number 4

### Existence of Dirac resonances in the semi-classical limit

Pages: 381 – 395

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

#### Authors

#### 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.