Affiliations: Département de Mathématiques et Applications, École Normale Supérieure, Paris, France E‐mails: Riccardo.Adami, [email protected] | Université de Paris 7, Laboratoire Jacques‐Louis Lions, France E‐mail: [email protected] | Dipartimento di Matematica Pura ed Applicata, Università di L'Aquila, Italy E‐mail: [email protected]
Note: [] Corresponding author. Current address: 45, rue d'Ulm, 75230 Paris cedex 05, France.
Note: [] One of us (R.A.) profited by a Marie Curie Fellowship, proposal n. MCFI‐2000‐01934, contract n. HPMF‐CT‐2000‐01102.
Note: [] Current address: 175, rue du Chevaleret, 750013 Paris, France.
Note: [] Current address: 45, rue d'Ulm, 75230 Paris cedex 05, France.
Note: [] Current address: Via Vetoio (Coppito 1), 67010 Coppito di L'Aquila (AQ), Italy.
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.
