Long-range scattering for a critical homogeneous type nonlinear Schr\"odinger equation with time-decaying harmonic potentials

TL;DR

This paper investigates the long-range scattering behavior of a critical nonlinear Schrödinger equation with time-decaying harmonic potentials, extending previous results to more general nonlinearities and improving conditions in lower dimensions.

Contribution

It generalizes prior work on gauge-invariant nonlinearities to include all homogeneous nonlinearities using Fourier series techniques and develops new factorization identities for the propagator.

Findings

Existence of solutions with logarithmic phase correction

Extension to general homogeneous nonlinearities

Improved regularity conditions in 2D and 3D

Abstract

This paper is concerned with the final state problem for the homogeneous type nonlinear Schr\"odinger equation with time-decaying harmonic potential. The nonlinearity has the critical order and is not necessarily the form of a polynomial. In the case of the gauge-invariant power-type nonlinearity, the first author proves that the equation admits a nontrivial solution that behaves like a free solution with a logarithmic phase correction in [22]. In this paper, we extend his result into the case with the general homogeneous nonlinearity by the technique due to the Fourier series expansion introduced by Masaki and the second author [26]. To adapt the argument in the aforementioned paper, we develop a factorization identity for the propagator and require a little stronger decay condition for the Fourier coefficients arising from the harmonic potential. Moreover, in two or three dimensions, we improve the regularity condition of the final data in [26, 28].