Well posedness of the nonlinear Schr\"odinger equation with isolated singularities

TL;DR

This paper investigates the well-posedness of the nonlinear Schrödinger equation with point interactions and power nonlinearities in 2D and 3D, establishing local and global existence results for various nonlinear powers.

Contribution

It provides new results on existence, uniqueness, and continuous dependence of solutions for NLS with singularities, extending understanding of singular solutions in higher dimensions.

Findings

Local existence and uniqueness of solutions in 2D and 3D.

Global existence for sub-cubic powers in 2D.

Restrictions on powers for well-posedness in 3D.

Abstract

We study the well posedness of the nonlinear Schr\"odinger (NLS) equation with a point interaction and power nonlinearity in dimension two and three. Behind the autonomous interest of the problem, this is a model of the evolution of so called singular solutions that are well known in the analysis of semilinear elliptic equations. We show that the Cauchy problem for the NLS considered enjoys local existence and uniqueness of strong (operator domain) solutions, and that the solutions depend continuously from initial data. In dimension two well posedness holds for any power nonlinearity and global existence is proved for powers below the cubic. In dimension three local and global well posedness are restricted to low powers.