Remarks on scattering matrices for Schrödinger operators with critically long-range perturbations

Abstract

We consider scattering matrix for Schrödinger-type operators on ${\mathbb{R}}^d$ with perturbation $V(x)=O(⟨x{⟩}^{-1})$ 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.

References in corpus (2)

• Microlocal resolvent estimates, revisited
• Long-range Scattering Matrix for Schrödinger-type Operators

Cited by in corpus (3)

• 0-th Order Pseudo-differential Operator on the Circle
• Long-range Scattering Matrix for Schrödinger-type Operators
• Stationary scattering theory for one-body Stark operators, I