Absence of zero resonances of massless Dirac operators (Q297521)

The author considers the massless Dirac operator \(H=\alpha D+Q(x), D=-i\nabla_x\) on the Hilbert space \(L^2({\mathbb R}^3,{\mathbb C}^4)\), where \(Q(x)\) is a \(4\times 4\) Hermitian matrix valued function such that for each component \(q_{jk}(x) (j,k=1,\dots,4)\) of \(Q(x)\) \(|q_{jk}(x)|\leq C\langle x\rangle^{-\rho}, \langle x\rangle =\sqrt{1+|x|^2} \), \(C<\infty, \rho>1\). It is proved that if \(f\in \langle x\rangle^{3/2}L^2({\mathbb R}^3,{\mathbb C}^4), Hf=0\) in the sense of distributions, then for any \(\mu < 1/2\), \(\langle x\rangle^\mu f\in H^1({\mathbb R}^3,{\mathbb C}^4)\), where \(H^1\) is the Sobolev space of order 1. In particular, there is no resonance for \(H\).