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.
Categories: math.SP