Derivation of the Time Dependent Two Dimensional Focusing NLS Equation

Abstract

In this paper, we present a microscopic derivation of the two-dimensional focusing cubic nonlinear Schrödinger 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 Schrödinger 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.

34 pages, v3: Corrected an error in the proof

References in corpus (4)

• A pseudoconformal compactification of the nonlinear Schrödinger equation and applications
• Norm approximation for many-body quantum dynamics: focusing case in low dimensions
• Derivation of the Time Dependent Gross Pitaevskii Equation with External Fields
• A simple proof of convergence to the Hartree dynamics in Sobolev trace norms

Cited by in corpus (8)

• Higher order corrections to the mean-field description of the dynamics of interacting bosons
• Beyond Bogoliubov Dynamics
• Derivation of the 1d Gross-Pitaevskii equation from the 3d quantum many-body dynamics of strongly confined bosons
• Derivation of the 1d NLS equation from the 3d quantum many-body dynamics of strongly confined bosons
• Derivation of the time dependent Gross-Pitaevskii equation for a class of non purely positive potentials
• Mean-field limits of particles in interaction with quantized radiation fields
• Derivation of the 2d Gross-Pitaevskii equation for strongly confined 3d bosons
• Scaling limits of bosonic ground states, from many-body to nonlinear Schr{ö}dinger