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

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 $V\in L_{\mathrm{loc}}^p({\mathbb{R}}^2)$, $p\in (2,\infty ]$, and $b\in L_{\mathrm{loc}}^q({\mathbb{R}}^2)$, $q\in (1,\infty ]$, 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.