Derivation of the Time Dependent Two Dimensional Focusing NLS Equation

TL;DR

This paper derives a two-dimensional focusing cubic nonlinear Schrödinger equation from an N-particle Bosonic system, establishing convergence of the reduced density matrix to the NLS solution under certain conditions.

Contribution

It provides a microscopic derivation of the 2D focusing NLS from many-body quantum dynamics with specific interaction potentials and initial states.

Findings

Convergence of reduced density matrices to NLS solution in Sobolev trace norm.

Validation of the derivation under various external potential conditions.

Extension of derivation to a class of bounded, compactly supported interactions.

Abstract

In this paper, we present a microscopic derivation of the two-dimensional focusing cubic nonlinear Schr\"odinger equation starting from an interacting $N$-particle system of Bosons. The interaction potential we consider is given by $W_\beta(x)=N^{-1+2 \beta}W(N^\beta x)$ for some bounded and compactly supported $W$. We assume the $N$-particle Hamiltonian fulfills stability of second kind. The class of initial wave functions is chosen such that the variance in energy is small. We then prove the convergence of the reduced density matrix corresponding to the exact time evolution to the projector onto the solution of the corresponding nonlinear Schr\"odinger equation in either Sobolev trace norm, if the external potential is in some $L^p$ space, $p \in ]2, \infty]$, or in trace norm, for more general external potentials.