# Localization of two-dimensional massless Dirac fermions in a magnetic quantum dot

### Martin Könenberg

Universität Wien, Austria### Edgardo Stockmeyer

Ludwig-Maximilians-Universität, München, Germany

## Abstract

We consider a two-dimensional massless Dirac operator $H$ in the presence of a perturbed homogeneous magnetic field $B=B_0+b$ and a scalar electric potential $V$. For $V\in L_{\rm loc}^p(\mathbb R^2)$, $p\in(2,\infty]$, and $b\in L_{\rm loc}^q(\mathbb R^2)$, $q\in(1,\infty]$, both decaying at infinity, we show that states in the discrete spectrum of $H$ are superexponentially localized. We establish the existence of such states between the zeroth and the first Landau level assuming that $V=0$. In addition, under the condition that $b$ is rotationally symmetric and that $V$ satisfies certain analyticity condition on the angular variable, we show that states belonging to the discrete spectrum of $H$ are Gaussian-like localized.

## Cite this article

Martin Könenberg, Edgardo Stockmeyer, Localization of two-dimensional massless Dirac fermions in a magnetic quantum dot. J. Spectr. Theory 2 (2012), no. 2, pp. 115–146

DOI 10.4171/JST/24