Quantum Dynamics with Mean Field Interactions a New Approach.pdf

Quantum Dynamics with Mean Field Interactions: a New Approach L´aszl´o Erd˝os∗ , Benjamin Schlein† 8 0 Institute of Mathematics, University of Munich, 0 2 Theresienstr. 39, D-80333 Munich, Germany r p April 23, 2008 A 3 2 ] Dedicated to J¨urg Fr¨ohlich on the occasion of his 60th birthday, h with admiration and gratitude p - h Abstract t a We propose a new approach for the study of the time evolution of a factorized N-particle m bosonic wave function with respect to a mean-field dynamics with a bounded interaction potential. [ The new technique, which is based on the control of the growth of the correlations among the particles, leads to quantitative bounds on the difference between the many-particle Schr¨odinger 1 v dynamics and the one-particle nonlinear Hartree dynamics. In particular the one-particle density 4 matrix associated with the solution to the N-particle Schr¨odinger equation is shown to converge to 7 the projection onto the one-dimensional subspace spanned by the solution to the Hartree equation 7 with a speed of convergence of order 1/N for all fixed times. 3 . 4 0 1 Introduction 8 0 : We consider a system of N interacting bosons in ν dimensions described on the Hilbert space HN = v 2 Nν 2 Nν L (R ,dx ,... dx ), the subspace of L (R ,dx ... dx ) consisting of permutation symmetric i s 1 N