An introduction to the two-dimensional Schr\"odinger equation with nonlinear point interactions

TL;DR

This paper introduces the two-dimensional nonlinear Schr"odinger equation with concentrated nonlinearities, discussing main challenges and recent results on well-posedness in this context.

Contribution

It provides an overview of the main difficulties and recent advances in understanding the well-posedness of the 2D NLSE with point interactions.

Findings

Discussion of main challenges in 2D NLSE with nonlinear point interactions

Summary of recent results on local and global well-posedness

Insights into the mathematical structure of concentrated nonlinearities

Abstract

We present an introduction to the nonlinear Schr\"odinger equation (NLSE) with concentrated nonlinearities in $\mathbb{R}^2$. Precisely, taking a cue from the linear problem, we sketch the main challenges and the typical difficulties that arise in the two-dimensional case, and mention some recent results obtained by the authors on local and global well-posedness.