Comment on the Uniqueness of the Ground State Solutions of a Fractional NLS with a Harmonic Potential

TL;DR

This paper discusses the application of a new unified method to prove the uniqueness of positive ground state solutions for fractional Schrödinger equations with harmonic potentials, an open problem in the field.

Contribution

The authors demonstrate how their general method can be applied to establish the uniqueness of ground states for a class of fractional Schrödinger equations with harmonic potential.

Findings

Their method confirms the uniqueness of ground states in this context.

They also explore existence, non-existence, and multiplicity of solutions.

Abstract

The uniqueness of the positive ground state solutions of fractional Shrodinger equations with a harmonic potential has not been covered by the breakthrough method developed in [1, 2]. It has remained an open question for years. [3] and [5] were quite recently able to prove the existence of the ground state solutions and to derive important properties but they failed to address the uniqueness. [3] left it as an open question, and [5] run numerical simulations that were in favor of the uniqueness. Very recently, the authors of this note developed a general and unified method to prove the uniqueness of the ground state solutions of a large class of variational problems, they also exhibited many examples to which their approach applies. After the publication of [4], some colleagues reached out to us to know whether our method applies to the fractional Shrodinger equation with a harmonic potential. In this short note, we will explain how it applies, and we will also state some existence/ non-existence and multiplicity solutions of the normalized