Decay and scattering of small solutions of pure power NLS in $\R$ with $p>3$ and with a potential

TL;DR

This paper proves decay and scattering for small solutions of a nonlinear Schrödinger equation with a potential in one dimension, extending understanding of long-term behavior in the presence of linear inhomogeneities.

Contribution

It establishes decay and scattering results for NLS with a potential and pure power nonlinearity in 1D, assuming small initial data and absence of discrete modes.

Findings

Solutions decay over time and scatter to linear solutions.

The presence of a Schwarz potential does not prevent scattering.

The proof uses operators commuting with the linearized equation.

Abstract

We prove decay and scattering of solutions of the Nonlinear Schr\"oding-er equation (NLS) in ${\mathbf R}$ with pure power nonlinearity with exponent $3<p<5$ when the initial datum is small in $\Sigma$ (bounded energy and variance), in the presence of a linear inhomogeneity represented by a linear potential which is a real valued Schwarz function. We assume absence of discrete modes. The proof is analogous to the one for the translation invariant equation. In particular we find appropriate operators commuting with the linearization.