Modified scattering for the cubic nonlinear Schr\"odinger equation with long-range potentials in one space dimension

TL;DR

This paper proves modified scattering for the cubic nonlinear Schrödinger equation with long-range potentials in one dimension, showing how solutions asymptotically resemble free solutions with phase corrections influenced by the potential and nonlinearity.

Contribution

It establishes the modified scattering in the energy space for equations with a broad class of long-range potentials, including those with negative eigenvalues, using a simple energy method.

Findings

Proves modified scattering in the energy space for the equation.

Handles a large class of long-range potentials, including negative ones.

Uses a simple energy method without relying on global Strichartz estimates.

Abstract

We consider the cubic nonlinear Schr\"odinger equation with long-range linear potentials in one space dimension, and prove the modified scattering in the energy space for the associated final state problem with a prescribed small asymptotic profile. Compared with the leading term of the free solution, the asymptotic profile has an additional phase correction depending both on the long-range part of the potential and on the nonlinear term. The proof is based on a simple energy method and does not rely on global-in-time Strichartz estimates for Schr\"odinger equations with linear potentials. In particular, the class of potentials to which our theorem applies is large enough to accommodate slowly decaying negative potentials so that the associated Schr\"odinger operators may have negative eigenvalues.