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

### Jean-François Bony

Université de Bordeaux, Talence, France### Nicolás Espinoza

University of Tokyo, Japan### Georgi Raikov

Pontificia Universidad Católica de Chile, Santiago, Chile

## 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 $b_{0}=0$, then generically dim Ker $H=∞$. If $b_{0}=0$, then for each $m∈N∪{∞}$, 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.

## Cite this article

Jean-François Bony, Nicolás Espinoza, Georgi Raikov, Spectral Properties of 2D Pauli Operators with Almost-Periodic Electromagnetic Fields. Publ. Res. Inst. Math. Sci. 55 (2019), no. 3, pp. 453–487

DOI 10.4171/PRIMS/55-3-1