Nonlinear Schrödinger equation with Coulomb potential

Abstract

In this paper, we study the Cauchy problem for the nonlinear Schrödinger equations with Coulomb potential $i{\partial }_{t}u+\Delta u+\frac{K}{|x|}u=\lambda |u{|}^{p-1}u$ with $1<p\le 5$ 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 Schrödinger 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. $K\le 0$. The argument is based on the interaction Morawetz-type inequalities and the equivalence of Sobolev norms.

28 pages

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Cited by in corpus (3)

• Two dimensional NLS ground states with attractive Coulomb potential and point interaction
• Long time behaviors for the inhomogeneous NLS with a potential in ${\mathbb{R}}^3$
• Orbital stability of standing waves for Schrödinger type equations with slowly decaying linear potential