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

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 Schrödinger 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 Schrödinger equation, and on the use of explicit low frequency cancellations as done by 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.

Asymptotic stability of a wide class of steady states for the Hartree equation for random fields

“…The first study of the asymptotic stability of Y is [7], which proved the case when d ≥ 4. [8] extended the previous stability result to d = 2, 3. [8] succeeded to deal with potentials w in a wide class that includes the delta measure.…”

“…[8] extended the previous stability result to d = 2, 3. [8] succeeded to deal with potentials w in a wide class that includes the delta measure. However, [7,8] needed somewhat strong conditions for f , in particular, f has to be smooth.…”

“…We briefly explain the background of (NLH) for convenience according to [16,12]; for more details, see [16,7,12] and [8]. The time evolution of wave functions of N fermions in d-dimensional space is described by the linear Schrödinger equation on R N d .…”

“…[8] succeeded to deal with potentials w in a wide class that includes the delta measure. However, [7,8] needed somewhat strong conditions for f , in particular, f has to be smooth. Thus, their results do not include Fermi gas at zero temperature, that is, |f (ξ)| 2 = χ {|ξ|≤µ} with µ > 0.…”

“…This formulation was introduced by de Suzzoni in [16]. It has infinitely many steady states, and [7,8] have studied its asymptotic stability. However, they required somewhat strong conditions for steady states, and the stability of Fermi gas at zero temperature, one of the most important steady states from the physics point of view, was left open.…”