Stability of Steady States for Hartree and Schrodinger Equations for Infinitely Many Particles

TL;DR

This paper proves scattering near steady states for a Hartree equation modeling infinitely many particles, extending previous results to higher dimensions and broader interaction potentials, including nonlinear Schrödinger equations.

Contribution

It extends scattering results for the Hartree equation to dimensions 2 and 3 with more general potentials, connecting to density matrix frameworks and employing advanced dispersive techniques.

Findings

Established scattering near steady states for the Hartree equation in 2D and 3D.

Extended the class of interaction potentials to include nonlinear Schrödinger equations.

Utilized dispersive techniques and low frequency cancellations for the analysis.

Abstract

We prove a scattering result near certain steady states for a Hartree equation for a random field. This equation describes the evolution of a system of infinitely many particles. It is an analogous formulation of the usual Hartree equation for density matrices. We treat dimensions 2 and 3, extending our previous result. We reach a large class of interaction potentials, which includes the nonlinear Schrodinger equation. This result has an incidence in the density matrices framework. The proof relies on dispersive techniques used for the study of scattering for the nonlinear Schrodinger equation, and on the use of explicit low frequency cancellations as in the work of Lewin and Sabin. To relate to density matrices, we use Strichartz estimates for orthonormal systems from Frank and Sabin, and Leibniz rules for integral operators.