Scattering for the nonlinear Schrodinger equation with a general one-dimensional confinement

TL;DR

This paper studies the scattering behavior of the defocusing nonlinear Schrödinger equation with a one-dimensional external potential that can grow rapidly, establishing existence, uniqueness, and wave operators under certain conditions.

Contribution

It proves existence and uniqueness of solutions and wave operators for the NLS with a general one-dimensional potential, extending scattering theory to more complex external fields.

Findings

Existence and uniqueness of solutions in a suitable functional framework.

Existence of wave operators for large nonlinearity powers.

Asymptotic completeness established through multiple approaches.

Abstract

We consider the defocusing nonlinear Schr{\"o}dinger equation in several space dimensions, in the presence of an external potential depending on only one space vari-able. This potential is bounded from below, and may grow arbitrarily fast at infinity. We prove existence and uniqueness in the associated Cauchy problem, in a suitable functional framework, as well as the existence of wave operators when the power of the nonlinearity is sufficiently large. Asymptotic completeness then stems from at least two approaches, which are briefly recalled.