Modified scattering for inhomogeneous nonlinear Schr\"odinger equations with and without inverse-square potential

TL;DR

This paper constructs modified wave operators for inhomogeneous nonlinear Schrödinger equations with critical long-range nonlinearities, including cases with inverse-square potential, demonstrating global solutions that match prescribed asymptotic profiles.

Contribution

It introduces a method to establish the existence of modified wave operators for both inhomogeneous and inverse-square potential cases, extending the understanding of long-range scattering in these equations.

Findings

Existence of unique global solutions with prescribed asymptotics.

Construction of modified wave operators for equations with inverse-square potential.

Results hold under radial symmetry in three dimensions.

Abstract

We consider the final state problem for the inhomogeneous nonlinear Schr\"odinger equation with a critical long-range nonlinearity. Given a prescribed asymptotic profile, which has a logarithmic phase correction compared with the free evolution, we construct a unique global solution which converges to the profile. As a consequence, the existence of modified wave operators for localized small scattering data is obtained. We also study the same problem for the case with the critical inverse-square potential under the radial symmetry. In particular, we construct the modified wave operators for the long-range nonlinear Schr\"odinger equation with the critical inverse-square potential in three space dimensions, under the radial symmetry.