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

Adami, R., Bardos, C., Golse, F., Teta, A. (2004). Towards a rigorous derivation of the cubic NLSE in dimension one. ASYMPTOTIC ANALYSIS, 40(2), 93-108.

Towards a rigorous derivation of the cubic NLSE in dimension one

ADAMI, RICCARDO;
2004

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
Articolo in rivista - Articolo scientifico
Cubic NLS Equation
English
93
108
16
Adami, R., Bardos, C., Golse, F., Teta, A. (2004). Towards a rigorous derivation of the cubic NLSE in dimension one. ASYMPTOTIC ANALYSIS, 40(2), 93-108.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10281/11565
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