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

## TL;DR

This paper rigorously derives the Hamiltonian structure of the nonlinear Schrödinger equation from quantum many-body systems, linking quantum and classical descriptions through geometric and symplectic methods.

## Contribution

It provides a rigorous derivation of the Hamiltonian and symplectic structure of NLS from quantum many-body systems, extending geometric methods to infinite-dimensional systems.

## Key findings

- Derived the Hamiltonian functional for NLS from quantum systems
- Constructed a weak symplectic structure for the NLS
- Connected quantum many-body dynamics with classical Hamiltonian systems

## 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.