Derivation of the Cubic Non-linear Schr\"odinger Equation from Quantum Dynamics of Many-Body Systems

TL;DR

This paper rigorously derives the cubic non-linear Schrödinger equation from the quantum dynamics of many-body Bose systems with short-range interactions, establishing a connection between microscopic quantum models and macroscopic nonlinear wave equations.

Contribution

It provides a rigorous proof of the convergence of many-body quantum dynamics to the cubic non-linear Schrödinger equation in three dimensions, extending to all k-particle density matrices.

Findings

One-particle density matrix converges to the cubic NLS solution.

Extension to k-particle density matrices for all positive integers k.

Validates the mean-field limit for Bose systems with short-range interactions.

Abstract

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 non-linear Schr\"odinger equation in a suitable scaling limit. The result is extended to $k$-particle density matrices for all positive integer $k$.