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arXiv · 1908.03847

A Rigorous Derivation of the Hamiltonian Structure for the Nonlinear Schr\"odinger Equation

Dana Mendelson·Andrea R. Nahmod·Nataša Pavlović·Matthew Rosenzweig·Gigliola Staffilani
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Abstract

We consider the cubic nonlinear Schr\"odinger 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 Schr\"odinger 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 for a system describing the evolution of finitely many indistinguishable classical particles.

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Dana Mendelson, Andrea R. Nahmod, Nataša Pavlović, Matthew Rosenzweig, Gigliola Staffilani. 2019-08-11. A Rigorous Derivation of the Hamiltonian Structure for the Nonlinear Schr\"odinger Equation. https://arxiv.org/abs/1908.03847

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