Zero modes in a system of Aharonov--Bohm solenoids on the Lobachevsky plane

TL;DR

This paper investigates zero modes of Pauli operators for a spin 1/2 particle on the Lobachevsky plane with Aharonov-Bohm solenoids, showing finite systems lack zero modes while infinite periodic systems can have them depending on flux.

Contribution

It demonstrates the existence and absence of zero modes in finite and infinite solenoid configurations on the Lobachevsky plane, revealing flux-dependent spectral properties.

Findings

Finite solenoid systems have no zero modes.

Infinite periodic systems can have zero modes depending on flux.

Zero modes depend on the flux carried by solenoids.

Abstract

We consider a spin 1/2 charged particle on the Lobachevsky plane subjected to a magnetic field corresponding to a discrete system of Aharonov-Bohm solenoids. Let $H^+$ and $H^-$ be the two components of the Pauli operator for spin up and down, respectively. We show that neither $H^+$ nor $H^-$ has a zero mode if the number of solenoids is finite. On the other hand, a construction is described of an infinite periodic system of solenoids for which either $H^+$ or $H^-$ has zero modes depending on the value of the flux carried by the solenoids.