Stability of steady states for Hartree and Schrödinger equations for infinitely many particles

Abstract

We prove a scattering result 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 [11]. 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 [24]. To relate to density matrices, we use Strichartz estimates for orthonormal systems from [16], and improved Leibniz rules for integral operators.