# Stability of equilibria for a Hartree equation for random fields

@article{Collot2018StabilityOE, title={Stability of equilibria for a Hartree equation for random fields}, author={Charles Collot and Anne-Sophie de Suzzoni}, journal={arXiv: Analysis of PDEs}, year={2018} }

We consider a Hartree equation for a random variable, which describes the temporal evolution of infinitely many Fermions. On the Euclidean space, this equation possesses equilibria which are not localised. We show their stability through a scattering result, with respect to localised perturbations in the defocusing case in high dimensions $d\geq 4$. This provides an analogue of the results of Lewin and Sabin \cite{LS2}, and of Chen, Hong and Pavlovi\'c \cite{CHP2} for the Hartree equation on…

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## References

SHOWING 1-10 OF 36 REFERENCES

The Hartree equation for infinitely many particles, II: Dispersion and scattering in 2D

- Mathematics, Physics
- 2014

We consider the nonlinear Hartree equation for an interacting gas containing infinitely many particles and we investigate the large-time stability of the stationary states of the form $f(-\Delta)$,…

THE HARTREE EQUATION FOR INFINITELY MANY PARTICLES

- 2013

We consider the nonlinear Hartree equation for an interacting gas containing infinitely many particles and we investigate the large-time stability of the stationary states of the form f(−∆),…

Scattering theory for the Gross-Pitaevskii equation in three dimensions

- Mathematics
- 2008

We study global behavior of small solutions of the Gross-Pitaevskii equation in three dimensions. We prove that disturbances from the constant equilibrium with small, localized energy, disperse for…

A Microscopic Derivation of the Time-Dependent Hartree-Fock Equation with Coulomb Two-Body Interaction

- Physics, Mathematics
- 2008

We study the dynamics of a Fermi gas with a Coulomb interaction potential, and show that, in a mean-field regime, the dynamics is described by the Hartree-Fock equation. This extends previous work of…

On the scattering problem for infinitely many fermions in dimensions $d\geq3$ at positive temperature

- Mathematics, Physics
- 2016

Mean–Field Evolution of Fermionic Systems

- Physics, Mathematics
- 2014

The mean field limit for systems of many fermions is naturally coupled with a semiclassical limit. This makes the analysis of the mean field regime much more involved, compared with bosonic systems.…

Global Well-Posedness of the NLS System for Infinitely Many Fermions

- Physics, Mathematics
- 2015

In this paper, we study the mean field quantum fluctuation dynamics for a system of infinitely many fermions with delta pair interactions in the vicinity of an equilibrium solution (the Fermi sea) at…

Mean field dynamics of fermions and the time-dependent Hartree-Fock equation

- Mathematics, Physics
- 2002

Mean-field Evolution of Fermionic Mixed States

- Physics, Mathematics
- 2014

In this paper we study the dynamics of fermionic mixed states in the mean-field regime. We consider initial states which are close to quasi-free states and prove that, under suitable assumptions on…

An equation on random variables and systems of fermions

- Mathematics, Physics
- 2015

In this paper, we consider an equation on random variables which can be reduced to the equation which describes the evolution of systems of fermions. We give some results of well-posedness for this…