Limiting absorption principle and radiation conditions for Schr\"odinger operators with long-range potentials

TL;DR

This paper extends fundamental spectral and scattering results for Schrödinger operators with long-range potentials, using elementary commutator techniques to refine and generalize previous theorems like Rellich's and Sommerfeld's.

Contribution

It introduces a new, elementary commutator-based approach to establish limiting absorption and radiation conditions for long-range Schrödinger operators, broadening the scope of existing results.

Findings

Proves Rellich's theorem for long-range potentials.

Establishes the limiting absorption principle in this context.

Provides a Sommerfeld uniqueness result.

Abstract

We show Rellich's theorem, the limiting absorption principle, and a Sommerfeld uniqueness result for a wide class of one-body Schr\"odinger operators with long-range potentials, extending and refining previously known results. Our general method is based on elementary commutator estimates, largely following the scheme developed recently by Ito and Skibsted.