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We establish that the Pauli operator describing a spin-1/2 two-dimensional quantum system with a singular magnetic field has, under certain conditions, an infinite-dimensional space of zero modes, possibly, both spin-up and spin-down, moreover there is a spectral gap separating the zero eigenvalue from the rest of the spectrum. In particular, infiniteness takes place if the field has infinite flux, which settles this previously unknown case of Aharonov-Casher theorem.

Abstract. The recent theorem by D. Luecking that finite rank Toeplitz-Bergman operators must be generated by a measure consisting of finitely many point masses is carried over to the manydimensional case.

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