Eigenvalue bounds for non-selfadjoint Dirac operators

TL;DR

This paper establishes eigenvalue bounds for non-selfadjoint Dirac operators with small potentials, showing eigenvalues are confined to specific regions in the complex plane or absent in the massless case.

Contribution

It provides new eigenvalue localization results for non-Hermitian Dirac operators using an abstract Birman-Schwinger principle and resolvent estimates.

Findings

Eigenvalues of massive Dirac operators are confined to two disks in the complex plane.

In the massless case, the spectrum remains unchanged and contains no discrete eigenvalues.

The results apply under smallness conditions on the potential in mixed norm spaces.

Abstract

In this work we prove that the eigenvalues of the $n$-dimensional massive Dirac operator $\mathscr{D}_0 + V$, $n\ge2$, 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 $L^1_{x_j} L^\infty_{\widehat{x}_j}$, 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 $\sigma(\mathscr{D}_0+V)=\sigma(\mathscr{D}_0)=\mathbb{R}$. 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 Schr\"odinger operator.