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Remarks on scattering matrices for Schrödinger operators with critically long-range perturbations (1804.05489v6)
Published 16 Apr 2018 in math-ph, math.AP, and math.MP
Abstract: We consider scattering matrix for Schr\"odinger-type operators on $\mathbb{R}d$ with perturbation $V(x)=O(\langle x\rangle{-1})$ as $|x|\to\infty$. We show that the scattering matrix (with time-independent modifiers) is a pseudodifferential operator, and analyze its spectrum. We present examples of which the spectrum of the scattering matrices have dense point spectrum, and absolutely continuous spectrum, respectively.