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We consider a spin 1/2 charged particle on the Lobachevsky plane subjected toa 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 anddown, respectively. We show that neither $H^+$ nor $H^-$ has a zero mode if thenumber of solenoids is finite. On the other hand, a construction is describedof an infinite periodic system of solenoids for which either $H^+$ or $H^-$ haszero modes depending on the value of the flux carried by the solenoids

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