Semiclassical limit of cubic nonlinear Schr\"odinger equations for mixed states

TL;DR

This paper investigates the semiclassical limit of cubic nonlinear Schrödinger equations for mixed states, establishing convergence to a singular Vlasov equation under specific regularity and stability conditions.

Contribution

It provides a rigorous justification of the semiclassical limit to a singular Vlasov equation for mixed states with finite Sobolev regularity, under a quantum Penrose stability condition.

Findings

Limit justified for small data in both focusing and defocusing cases

Applicable to perturbations of Maxwellian-like profiles in the defocusing case

Stability condition always satisfied for small data

Abstract

In this work, we study the semiclassical limit of cubic Nonlinear Schr\"odinger equations for mixed states. We justify the limit to a singular Vlasov equation (in which the force field is proportional to the gradient of the density), for data with finite Sobolev regularity whose velocity profiles satisfy a quantum Penrose stability condition. This latter condition is always satisfied for small data (with a smallness condition independent of the semiclassical parameter) both in the focusing and the defocusing case, and for small perturbations of a large class of physically relevant examples in the defocusing case, such as local Maxwellian-like profiles.