A rigorous derivation of the Hamiltonian structure for the nonlinear Schrödinger equation

Abstract

© 2020 Elsevier Inc. We consider the cubic nonlinear Schrödinger equation (NLS) in any spatial dimension, which is a well-known example of an infinite-dimensional Hamiltonian system. Inspired by the knowledge that the NLS is an effective equation for a system of interacting bosons as the particle number tends to infinity, we provide a derivation of the Hamiltonian structure, which is comprised of both a Hamiltonian functional and a weak symplectic structure, for the nonlinear Schrödinger equation from quantum many-body systems. Our geometric constructions are based on a quantized version of the Poisson structure introduced by Marsden, Morrison and Weinstein [24] for a system describing the evolution of finitely many indistinguishable classical particles.