Bound states of two-dimensional Schr\"{o}dinger-Newton equations

TL;DR

This paper establishes the existence and uniqueness of ground states and angular excitations for 2D Schrödinger-Newton equations, deriving a sharp inequality from the minimization problem.

Contribution

It provides the first rigorous proof of ground state properties and a new sharp inequality for the 2D Schrödinger-Newton system.

Findings

Proved existence and uniqueness of ground states.

Derived a sharp logarithmic Hardy-Littlewood-Sobolev inequality.

Characterized angular excitations of the system.

Abstract

We prove an existence and uniqueness result for ground states and for purely angular excitations of two-dimensional Schr\"{o}dinger-Newton equations. From the minimization problem for ground states we obtain a sharp version of a logarithmic Hardy-Littlewood-Sobolev type inequality.