The Rigorous Derivation of the 2D Cubic Focusing NLS from Quantum Many-body Evolution

Abstract

We consider a 2D time-dependent quantum system of $N$-bosons with harmonic external confining and \emph{attractive} interparticle interaction in the Gross-Pitaevskii scaling. We derive stability of matter type estimates showing that the $k$-th power of the energy controls the $H^1$ Sobolev norm of the solution over $k$-particles. This estimate is new and more difficult for attractive interactions than repulsive interactions. For the proof, we use a version of the finite-dimensional quantum di Finetti theorem from [49]. A high particle-number averaging effect is at play in the proof, which is not needed for the corresponding estimate in the repulsive case. This a priori bound allows us to prove that the corresponding BBGKY hierarchy converges to the GP limit as was done in many previous works treating the case of repulsive interactions. As a result, we obtain that the \emph{focusing} nonlinear Schrödinger equation is the mean-field limit of the 2D time-dependent quantum many-body system with attractive interatomic interaction and asymptotically factorized initial data. An assumption on the size of the $L^1$-norm of the interatomic interaction potential is needed that corresponds to the sharp constant in the 2D Gagliardo-Nirenberg inequality though the inequality is not directly relevant because we are dealing with a trace instead of a power.

References in corpus (5)

• One-and-a-half quantum de Finetti theorems
• On the Mean-Field Limit of Bosons with Coulomb Two-Body Interaction
• On the uniqueness of solutions to the Gross-Pitaevskii hierarchy
• Correlation structures, Many-body Scattering Processes and the Derivation of the Gross-Pitaevskii Hierarchy
• On the unconditional uniqueness of solutions to the infinite radial Chern-Simons-Schrödinger hierarchy

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