DERIVATION OF THE TWO-DIMENSIONAL NONLINEAR SCHRÖDINGER EQUATION FROM MANY BODY QUANTUM DYNAMICS By KAY KIRKPATRICK, BENJAMIN SCHLEIN, and GIGLIOLA STAFFILANI
نویسندگان
چکیده
We derive rigorously, for both R2 and [−L, L]×2, the cubic nonlinear Schrödinger equation in a suitable scaling limit from the two-dimensional many-body Bose systems with short-scale repulsive pair interactions. We first prove convergence of the solution of the BBGKY hierarchy, corresponding to the many-body systems, to a solution of the infinite Gross-Pitaevskii hierarchy, corresponding to the cubic NLS; and then we prove uniqueness for the infinite hierarchy, which requires number-theoretical techniques in the periodic case.
منابع مشابه
Derivation of the two-dimensional nonlinear Schrödinger equation from many body quantum dynamics
We derive rigorously, for both R and [−L,L], the cubic nonlinear Schrödinger equation in a suitable scaling limit from the two-dimensional many-body Bose systems with short-scale repulsive pair interactions. We first prove convergence of the solution of the BBGKY hierarchy, corresponding to the many-body systems, to a solution of the infinite Gross-Pitaevskii hierarchy, corresponding to the cub...
متن کاملDerivation of the two-dimensional nonlinear Schrodinger equation from many body quantum dynamics Citation
We derive rigorously, for both R and [−L,L], the cubic nonlinear Schrödinger equation in a suitable scaling limit from the two-dimensional many-body Bose systems with short-scale repulsive pair interactions. We first prove convergence of the solution of the BBGKY hierarchy, corresponding to the many-body systems, to a solution of the infinite Gross-Pitaevskii hierarchy, corresponding to the cub...
متن کاملThe Gross-pitaevskii Hierarchy on General Rectangular Tori
In this work, we study the Gross-Pitaevskii hierarchy on general –rational and irrational– rectangular tori of dimension two and three. This is a system of infinitely many linear partial differential equations which arises in the rigorous derivation of the nonlinear Schrödinger equation. We prove a conditional uniqueness result for the hierarchy. In two dimensions, this result allows us to obta...
متن کاملDerivation of the Cubic Non-linear Schrödinger Equation from Quantum Dynamics of Many-Body Systems
We prove rigorously that the one-particle density matrix of three dimensional interacting Bose systems with a short-scale repulsive pair interaction converges to the solution of the cubic nonlinear Schrödinger equation in a suitable scaling limit. The result is extended to k-particle density matrices for all positive integer k. AMS Classification Number (2000): 35Q55, 81Q15, 81T18, 81V70. Runni...
متن کاملThe Quintic Nls as the Mean Field Limit of a Boson Gas with Three-body Interactions
We investigate the dynamics of a boson gas with three-body interactions in dimensions d = 1, 2. We prove that in the limit where the particle number N tends to infinity, the BBGKY hierarchy of k-particle marginals converges to a limiting (Gross-Pitaevskii (GP)) hierarchy for which we prove existence and uniqueness of solutions. The solutions of the GP hierarchy are shown to be determined by sol...
متن کامل