The Dirac operator with mass $m_0\ge 0$: non-existence of zero modes and of threshold eigenvalues

Abstract

A simple global condition on the potential is given which excludes zero modes of the massless Dirac operator. As far as local conditions at infinity are concerned, it is shown that at energy zero the Dirac equation without mass term has no non-trivial $L^2$-solutions at infinity for potentials which are either very slowly varying or decaying at most like $r^{-s}$ with $s\in (0,1)$. When a mass term is present, it is proved that at the thresholds there are again no such solutions when the potential decays at most like $r^{-s}$ with $s\in (0,2)$. In both situations the decay rate is optimal.

A correction to this paper is available.