Eigenvalue bounds for non-selfadjoint Dirac operators

Abstract

In this work we prove that the eigenvalues of the $n$-dimensional massive Dirac operator ${\mathcal{D}}_0+V$, $n\ge 2$, perturbed by a possibly non-Hermitian potential $V$, are localized in the union of two disjoint disks of the complex plane, provided that $V$ is sufficiently small with respect to the mixed norms $L_{x_j}^1{L}_{{\hat{x}}_j}^{\infty }$, for $j\in \{1,\dots ,n\}$. In the massless case, we prove instead that the discrete spectrum is empty under the same smallness assumption on $V$, and in particular the spectrum is the same of the unperturbed operator, namely $\sigma ({\mathcal{D}}_0+V)=\sigma ({\mathcal{D}}_0)=\mathbb{R}$. The main tools we employ are an abstract version of the Birman-Schwinger principle, which include also the study of embedded eigenvalues, and suitable resolvent estimates for the Schrödinger operator.

20 pages