Scattering Theory in Quantum Mechanical Problems

TL;DR

This paper provides an overview of the mathematical foundations of scattering theory in quantum mechanics, covering two- and three-particle systems, and discusses recent developments in long-range interactions.

Contribution

It introduces both time-dependent and stationary approaches to scattering theory and addresses the complex problem of asymptotic completeness for three-particle systems, including new scattering channels.

Findings

Describes wave and scattering operators and matrices for Schrödinger equations.

Explains approaches for three interacting particles and asymptotic completeness.

Discusses new scattering channels for long-range interactions.

Abstract

The aim of the lecture is to briefly describe the mathematical background of scattering theory for two- and three-particle quantum systems. We discuss basic objects of the theory: wave and scattering operators and the corresponding scattering matrix and illustrate them on the example of the Schr\"odinger equation. Our goal is to present time-dependent and stationary approaches and to describe the underlying mathematical methods. We also give a sketch of scattering theory for three interacting quantum particles including a difficult problem of the asymptotic completeness of scattering channels. Along with traditional results, we discuss new scattering channels arising for long-range pair interactions.