We consider the cubic nonlinear Schrodinger 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 Schrodinger 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.

中文翻译：

---

非线性薛定谔方程哈密顿结构的严格推导

我们在任何空间维度上考虑三次非线性薛定谔方程 (NLS)，这是无限维哈密顿系统的一个众所周知的例子。由于粒子数趋于无穷大，NLS 是相互作用玻色子系统的有效方程这一知识的启发，我们提供了哈密顿结构的推导，该结构由哈密顿泛函和弱辛结构组成，对于来自量子多体系统的非线性薛定谔方程。我们的几何构造基于由 Marsden、Morrison 和 Weinstein 引入的泊松结构的量化版本，用于描述有限许多无法区分的经典粒子的演化的系统。