Collapsing Estimates and the Rigorous Derivation of the 2d Cubic Nonlinear Schr\"odinger Equation with Anisotropic Switchable Quadratic Traps

TL;DR

This paper rigorously derives the 2d cubic nonlinear Schrödinger equation with anisotropic switchable quadratic traps from many-body Schrödinger equations, extending collapsing estimates and establishing hierarchy uniqueness in 3d.

Contribution

It extends collapsing estimates and applies a generalized lens transform to derive the 2d cubic NLS with anisotropic traps and proves hierarchy uniqueness in 3d without factorized initial data.

Findings

Derived 2d cubic NLS with anisotropic traps rigorously.

Extended collapsing estimates in existing frameworks.

Established hierarchy uniqueness in 3d without initial data factorization.

Abstract

We consider the 2d and 3d many body Schr\"odinger equations in the presence of anisotropic switchable quadratic traps. We extend and improve the collapsing estimates in Klainerman-Machedon [24] and Kirkpatrick-Schlein-Staffilani [23]. Together with an anisotropic version of the generalized lens transform in Carles [3], we derive rigorously the cubic NLS with anisotropic switchable quadratic traps in 2d through a modified Elgart-Erd\"os-Schlein-Yau procedure. For the 3d case, we establish the uniqueness of the corresponding Gross-Pitaevskii hierarchy without the assumption of factorized initial data.