Well posedness of the nonlinear Schr\"odinger equation with isolated singularities
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.
Keywords
Cite
@article{arxiv.2012.12391,
title = {Well posedness of the nonlinear Schr\"odinger equation with isolated singularities},
author = {Claudio Cacciapuoti and Domenico Finco and Diego Noja},
journal= {arXiv preprint arXiv:2012.12391},
year = {2021}
}
Comments
Minor corrections; typos fixed; references added