Spectral Properties of 2D Pauli Operators with Almost-Periodic Electromagnetic Fields

Abstract

We consider a 2D Pauli operator with almost-periodic field $b$ and electric potential $V$. First, we study the ergodic properties of $H$ and show, in particular, that its discrete spectrum is empty if there exists a magnetic potential which generates the magnetic field $b-b_0$, where $b_0$ is the mean value of $b$. Next, we assume that $V=0$, and investigate the zero modes of $H$. As expected, if $b_0\ne 0$, then generically dim Ker $H=\infty$. If $b_0=0$, then for each $m\in \mathbb{N}\cup \{\infty \}$, we construct an almost-periodic $b$ such that dim Ker $H=m$. This construction depends strongly on results concerning the asymptotic behavior of Dirichlet series, also obtained in the present article.