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

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.