Almost surely smoothed scattering for cubic NLS

TL;DR

This paper proves that solutions to the cubic nonlinear Schrödinger equation in dimensions 2 to 4, with randomized low-regularity initial data, scatter almost surely and exhibit smoothing properties with quantifiable convergence rates.

Contribution

It introduces a probabilistic approach to establish scattering and smoothing for cubic NLS at low regularity, advancing understanding of solution behavior in these regimes.

Findings

Solutions scatter almost surely with randomized initial data.

Smoothing properties of the scattering operator are established.

Convergence rates of the scattering process are quantified.

Abstract

We consider cubic NLS in dimensions 2, 3, 4 and we prove that almost surely solutions with randomized initial data at low regularity scatter. Moreover, we establish some smoothing properties of the associated scattering operator and precise the rate of convergence.