- 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