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

Abstract

We consider the Pauli operator in ${\mathbb{R}}^3$ for magnetic fields in $L^{3/2}$ that decay at infinity as $|x{|}^{-2-\beta }$ 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 $\delta (\mathbf{B})$, defined below, being equal to zero. Complementing a result from [Balinsky, Evans, Lewis (2001)], this implies that for the class of magnetic fields considered, Sobolev, Hardy and CLR inequalities hold whenever the magnetic field has no zero mode.

9 pages