# Wave operators on Sobolev spaces

## TL;DR

This paper establishes a general condition for the existence and completeness of wave operators on Sobolev spaces, with applications to Schrödinger operators and nonlinear Schrödinger equations.

## Contribution

It introduces a simple abstract criterion linking wave operators on Sobolev spaces to classical wave operators, expanding scattering theory tools.

## Key findings

- Abstract condition for wave operators on Sobolev spaces
- Application to Schrödinger operators with potentials
- Extension to nonlinear Schrödinger equations

## Abstract

We provide a simple sufficient condition in an abstract framework to deduce the existence and completeness of wave operators (resp. modified wave operators) on Sobolev spaces from the existence and completeness of the usual wave operators (resp. modified wave operators). We then give some examples of Schr\"odinger operators to which our abstract result applies. An application to scattering theory for the nonlinear Schr\"odinger equation with a potential is also given.