Spectral properties of 2D Pauli operators with almost periodic electromagnetic fields

TL;DR

This paper investigates the spectral and zero mode properties of 2D Pauli operators with almost periodic electromagnetic fields, revealing conditions for the absence of discrete spectrum and constructing operators with prescribed zero mode dimensions.

Contribution

It provides new insights into the ergodic properties and zero modes of 2D Pauli operators with almost periodic fields, including explicit constructions based on Dirichlet series asymptotics.

Findings

Discrete spectrum is empty under certain almost periodic magnetic potentials.

Infinite zero modes occur when the mean magnetic field is non-zero.

Constructs almost periodic fields with any prescribed finite or infinite zero mode dimension.

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 an almost periodic magnetic potential which generates the magnetic field $b - b_{0}$, $b_{0}$ being the mean value of $b$. Next, we assume that $V = 0$, and investigate the zero modes of $H$. As expected, if $b_{0} \neq 0$, then generically $\operatorname{dim} \operatorname{Ker} H = \infty$. If $b_{0} = 0$, then for each $m \in {\mathbb N} \cup \{ \infty \}$, we construct almost periodic $b$ such that $\operatorname{dim} \operatorname{Ker} H = m$. This construction depends strongly on results concerning the asymptotic behavior of Dirichlet series, also obtained in the present article.