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.

arXiv comment9 pages

References in corpus (3)

• The kernel of Dirac operators on ${\S }^3$ and ${\mathbb{R}}^3$
• Zero modes of the Dirac operator in three dimensions
• Degeneracy of zero modes of the Dirac operator in three dimensions

Cited by in corpus (1)

• On Absence of Threshold Resonances for Schrodinger and Dirac Operators