Derivation of the nonlinear Schrödinger equation with a general nonlinearity and Gross–Pitaevskii hierarchy in one and two dimensions

Abstract

In this paper, we investigate the quantum many-body dynamics with a linear combination of many-body interactions. We derive rigorously the nonlinear Schrödinger equation with a general nonlinearity as the mean-field limit of this model. Due to the complex interaction structure, we establish a new energy estimate for 0<β<1(m−1)d, which is efficient to handle the case of many-body interactions and allows us to obtain the mean-field approximation on longer length scales than the previous result in the work of Xie [Differ. Integr. Equations 28, 455–504 (2015)].

Topics & Concepts