Nonlinear Schr\"odinger equation with Coulomb potential

TL;DR

This paper investigates the nonlinear Schrödinger equation with Coulomb potential, establishing global existence for attractive potentials and scattering for repulsive ones, using Morawetz inequalities and Sobolev norm equivalence.

Contribution

It provides new results on existence and scattering for NLS with Coulomb potential, highlighting the potential's influence on solution behavior.

Findings

Global existence for attractive Coulomb potential ($K>0$).

Scattering results for repulsive Coulomb potential ($K extless=0$).

Use of interaction Morawetz inequalities and Sobolev norm techniques.

Abstract

In this paper, we study the Cauchy problem for the nonlinear Schr\"odinger equations with Coulomb potential $i\partial_tu+\Delta u+\frac{K}{|x|}u=\lambda|u|^{p-1}u$ with $1<p\leq5$ on $\mathbb{R}^3$. We mainly consider the influence of the long range potential $K|x|^{-1}$ on the existence theory and scattering theory for nonlinear Schr\"odinger equation. In particular, we prove the global existence when the Coulomb potential is attractive, i.e. $K>0$ and scattering theory when the Coulomb potential is repulsive i.e. $K\leq0$. The argument is based on the interaction Morawetz-type inequalities and the equivalence of Sobolev norms.