03-347 R.Adami, C.Bardos, F.Golse, A.Teta

Towards a rigorous derivation of the cubic NLSE in dimension one (52K, LATeX 2e) Jul 24, 03

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Abstract. We consider a system of N particles in dimension one, interacting through a zero-range repulsive potential whose strength is proportional to 1/N. We construct the finite and the infinite Schroedinger 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 Schr\"odinger 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.

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