Low energy spectral and scattering theory for relativistic Schrödinger operators (Q297513)

Spectral and scattering theory at low energy is investigated for the relativistic Schrödinger operator \(H=\sqrt{-\Delta}+V\) on \(L^2(\mathbb{R}^3)\). \(H\) is regarded as a perturbation of \(H_0=\sqrt{-\Delta}\). The potential \(V\) is real, measurable and \(V(x) = O((1+|x|^2)^{-\sigma/2}\) for \(\sigma>1\). The authors show that for a generic class (in \(L^3\)) of potentials, 0 is not an eigenvalue of \(H\) and if 0 is not an eigenvalue then \(H\) does not have a zero energy resonance and 0 cannot be an accumulation point of positive eigenvalues. They further show that, when \(\sigma>3\), the scattering matrix \(S(\lambda)\) converges in operator norm to the identity operator in the low energy limit (\(\lambda \to 0\)). Conditions are also derived that assure \(H\) has purely absolutely continuous spectrum on \([0,\infty)\). A formula for \(e^{itH_0}\) analogous to that known for \(e^{it(-\Delta)}\) is established in an Appendix.