A Remark on the Mourre Theory for Two Body Schr\"odinger operators

TL;DR

This paper adapts Mourre theory for two-body Schrödinger operators by using difference conditions on the potential, modifying the conjugate operator, and demonstrating applicability through examples.

Contribution

It introduces a new approach to Mourre theory using difference conditions instead of derivative conditions, expanding its applicability.

Findings

Successfully applies the modified Mourre theory to specific examples.

Shows the effectiveness of difference conditions on potentials.

Provides a framework for broader application of Mourre theory.

Abstract

On this short note, we apply the Mourre theory of the limiting absorption with {\it difference} type conditions on the potential, instead of conditions on the derivatives. In order that we modify the definition of the conjugate operator, and we apply the standard abstract Mourre theory. We also discuss examples to which the method applies.