# N Particle Limits and Derivation of the Self Consistent Non-linear Schrödinger Equation

Tuesday, May 23, 2000 - 9:30am - 10:10am

Keller 3-180

Claude Bardos (Université de Paris VII (Denis Diderot))

This talk is report on joint work with F. Golse and N. Mauser and is devoted to the problem of the N particle limit.

Continuing the talk of Norbert Mauser in the previous workshop, it focus mostly on the derivation of the N-particle Schrödinger equation in the time dependent case.

It emphazises the role of a so-called “finite Schrödinger hierarchy and of a limiting (infinite) “Schrödinger hierarchy. Convergence of solutions of the first to solutions of the second is established by using “physical relevant estimates (L2 and energy conservation) under very general assumptions on the interaction potential, including in particular the Coulomb potential. In the case of bounded potentials, a stability theorem for the infinite Schrödinger hierarchy is proved, based on Spohn's idea of using the trace norm and elementary techniques pertaining to the abstract Cauchy-Kowalewskaya theorem. The core of this program is to prove that if the limiting N-particle distribution function is factorized at time t=0, it remains factorized for all later times. The stability result above is a slight improvement of Spohn's uniqueness result for the infinite Schrödinger hierarchy, obtained 20 years ago, and which remains to this date the only rigorous derivation from first principles of the (nonlinear) 1-particle Schrödinger with self-consistent potential.

Continuing the talk of Norbert Mauser in the previous workshop, it focus mostly on the derivation of the N-particle Schrödinger equation in the time dependent case.

It emphazises the role of a so-called “finite Schrödinger hierarchy and of a limiting (infinite) “Schrödinger hierarchy. Convergence of solutions of the first to solutions of the second is established by using “physical relevant estimates (L2 and energy conservation) under very general assumptions on the interaction potential, including in particular the Coulomb potential. In the case of bounded potentials, a stability theorem for the infinite Schrödinger hierarchy is proved, based on Spohn's idea of using the trace norm and elementary techniques pertaining to the abstract Cauchy-Kowalewskaya theorem. The core of this program is to prove that if the limiting N-particle distribution function is factorized at time t=0, it remains factorized for all later times. The stability result above is a slight improvement of Spohn's uniqueness result for the infinite Schrödinger hierarchy, obtained 20 years ago, and which remains to this date the only rigorous derivation from first principles of the (nonlinear) 1-particle Schrödinger with self-consistent potential.