Derivation of a nonlinear Schrödinger equation with a general power-type nonlinearity in $$d=1,2$$.(English)Zbl 1363.35358

Summary: 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 [L. Erdős et al., Invent. Math. 167, No. 3, 515–614 (2007; Zbl 1123.35066)], [K. Kirkpatrick et al., Am. J. Math. 133, No. 1, 91–130 (2011; Zbl 1208.81080)] and [T. Chen and N. Pavlović, J. Funct. Anal. 260, No. 4, 959–997 (2011; Zbl 1213.35368)]. For the uniqueness part, we expand the procedure in [S. Klainerman and M. Machedon, Commun. Math. Phys. 279, No. 1, 169–185 (2008; Zbl 1147.37034)] 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 [S. Klainerman and M. Machedon, loc. cit.] is removed in our proof.

MSC:

 35Q55 NLS equations (nonlinear Schrödinger equations) 81V70 Many-body theory; quantum Hall effect
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