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

Abstract

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 $\alpha \cdot (-i\nabla -\text{A})+m\beta$ 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.

References in corpus (3)

• Lieb-Thirring inequalities for Schrödinger operators with complex-valued potentials
• Variation of discrete spectra for non-selfadjoint perturbations of selfadjoint operators
• Number of eigenvalues for a class of non-selfadjoint Schrödinger operators

Cited by in corpus (1)

• Location of eigenvalues of non-self-adjoint discrete Dirac operators