A criterion for the existence of zero modes for the Pauli operator with fastly decaying fields

摘要

We consider the Pauli operator in R-3 for magnetic fields in L-3/2 that decay at infinity as vertical bar x vertical bar(-2-beta) with beta > 0. In this case, we are able to prove that the existence of a zero mode for this operator is equivalent to a quantity delta(B), defined below, being equal to zero. Complementing a result from Balinsky et al. [J. Phys. A: Math. Gen. 34, L19-L23 (2001)], this implies that for the class of magnetic fields considered, Sobolev, Hardy, and Cwikel, Lieb, Rosenblum (CLR) inequalities hold whenever the magnetic field has no zero mode. (C) 2015 AIP Publishing LLC.