Spectrum of the perturbed Landau-Dirac operator

TL;DR

This paper investigates how compactly supported perturbations affect the distribution of discrete eigenvalues near Landau-Dirac levels for the Dirac operator with a magnetic field, revealing new phenomena related to eigenvalue finiteness.

Contribution

It provides a three-term asymptotic formula for eigenvalue counting near Landau-Dirac levels and explores the effects of variable sign perturbations on eigenvalue finiteness.

Findings

Asymptotic eigenvalue distribution formula established

Finiteness or infiniteness of eigenvalues depends on perturbation sign and structure

New phenomena identified in variable sign perturbation cases

Abstract

In this article, we consider the Dirac operator with constant magnetic field in $\mathbb R^2$. Its spectrum consists of eigenvalues of infinite multiplicities, known as the Landau-Dirac levels. Under compactly supported perturbations, we study the distribution of the discrete eigenvalues near each Landau-Dirac level. Similarly to the Landau (Schr\"odinger) operator, we demonstrate that a three-terms 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.