Derivation of a nonlinear Schrödinger equation with a general power-type nonlinearity in $d=1,2$

Abstract

In this paper, we study the derivation of a certain type of NLS from many-body interactions of bosonic particles in $d=1,2$. We consider a model with a finite linear combination of $n$-body interactions and obtain that the $k$-particle marginal density of the BBGKY hierarchy converges when particle number goes to infinity. Moreover, the limit solves a corresponding infinite Gross-Pitaevskii hierarchy. We prove the uniqueness of factorized solution to the Gross-Pitaevskii hierarchy based on a priori space time estimates. The convergence is established by adapting the arguments originated or developed in [6], [15] and [2]. For the uniqueness part, we expand the procedure in [16] by introducing a different board game argument to handle the factorial in the number of terms from Duhamel expansion. The space time bound assumption in [16] is removed in our proof.