The Dirac Operator with Mass m_0 >= 0 : Non-Existence of Zero Modes and of Threshold Eigenvalues

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.

2010 Mathematics Subject Classification: Primary 35P15; Secondary 81Q10.

Keywords and Phrases: Dirac operators, virial theorem, threshold eigenvalue, zero mode.

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