Weak coupling limit of the $N$-particle Schrödinger equation

Abstract

This work is devoted to the derivation of a nonlinear 1-particle equation from a linear AT-particle Schrodinger equation in the time dependent case. It emphazises the role of a so-called "finite Schrodinger hierarchy" and of a limiting (infinite) "Schrodinger hierarchy". Convergence of solutions of the first to solutions of the second is established by using "physically relevant" estimates (L 2 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 Schrodinger 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 AT-particle distribution function is factorized at time t = 0, it remains factorized for all later times.