A criterion for the existence of non-real eigenvalues for a Dirac operator

The aim of this work is to explore the discrete spectrum generated by complex perturbations in $L^{2}(\mathbb{R}^3,\mathbb{C}^4)$ of the $3d$ Dirac operator $\alpha \cdot (-i\nabla - \textbf{A}) + m \beta$ with variable magnetic field. Here, $\alpha := (\alpha_1,\alpha_2,\alpha_3)$ and $\beta$ are $4 \times 4$ Dirac matrices, and $m > 0$ is the mass of a particle. We give a simple criterion for the potentials to generate discrete spectrum near $\pm m$. In the case of creation of non-real eigenvalues, this criterion gives also their location.