Microscopic derivation of a Schr\"odinger equation in dimension one with a nonlinear point interaction

TL;DR

This paper rigorously derives a one-dimensional nonlinear Schrödinger equation with a point interaction from many-body bosonic dynamics involving an impurity, providing a mathematical foundation for the nonlinear delta model used in physics.

Contribution

It presents the first rigorous derivation of the nonlinear delta model from many-body quantum dynamics with an impurity in one dimension.

Findings

Proves propagation of chaos for the system.

Establishes convergence of one-particle density operators.

Provides superexponential estimates of fluctuations.

Abstract

We derive an effective equation for the dynamics of many identical bosons in dimension one in the presence of a tiny impurity. The interaction between every pair of bosons is mediated by the impurity through a positive three-body potential. Assuming a simultaneous mean-field and short-range scaling with the short-range proceeding slower than the mean-field, and choosing an initial fully condensed state, we prove propagation of chaos and obtain an effective one-particle Schr\"odinger equation with a defocusing nonlinearity concentrated at a point. More precisely, we prove the convergence of one-particle density operators in the trace-class topology and estimate the fluctuations as superexponential. This is the first derivation of the so-called nonlinear delta model, widely investigated in the last decades, as a phenomenological model for several physical phenomena.