Spectrum of the Perturbed Landau-Dirac Operator

In this article, we consider the Dirac operator with constant magnetic field in R2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb R}<^>2$$\end{document}. Its spectrum consists of eigenvalues of infinite multiplicities, known as the Landau-Dirac levels. Under compactly supported perturbations of the electric and magnetic potentials, we study the distribution of the discrete eigenvalues near each Landau-Dirac level. Similar to the Landau (Schr & ouml;dinger) operator, we demonstrate that a three-term asymptotic formula holds for the eigenvalue counting function. One of the main novelties of this work is the treatment of some perturbations of variable sign. In this context, we explore some remarkable phenomena related to the finiteness or infiniteness of the discrete eigenvalues, which depend on the interplay of the different terms in the matrix perturbation.