Towards a rigorous derivation of the cubic NLSE in dimension one

Abstract

We consider a system of N particles in dimension one, interacting through a zero‐range repulsive potential whose strength is proportional to N⁻¹. We construct the finite and the infinite Schrödinger hierarchies for the reduced density matrices of subsystems with n particles. We show that the solution of the finite hierarchy converges in a suitable sense to a solution of the infinite one. Besides, the infinite hierarchy is solved by a factorized state, built as a tensor product of many identical one‐particle wave functions which fulfil the cubic nonlinear Schrödinger equation. Therefore, choosing a factorized initial datum and assuming propagation of chaos, we provide a derivation for the cubic NLSE. The result, achieved with operator‐analysis techniques, can be considered as a first step towards a rigorous deduction of the Gross–Pitaevskii equation in dimension one. The problem of proving propagation of chaos is left untouched.