Uniqueness and stability of normalized ground states for Hartree equation with a harmonic potential

TL;DR

This paper investigates the existence, uniqueness, and stability of normalized ground states for the Hartree equation with a harmonic potential, providing a comprehensive analysis of their dynamic properties.

Contribution

It establishes the existence for any prescribed mass, proves uniqueness via convexity, and demonstrates orbital stability of the ground states.

Findings

Normalized ground states exist for any prescribed mass.

Uniqueness is confirmed through convexity of the energy functional.

Orbital stability of ground states is proven.

Abstract

The dynamic properties of normalized ground states for the Hartree equation with a harmonic potential are addressed. The existence of normalized ground state for any prescribed mass is confirmed according to mass-energy constrained variational approach. The uniqueness is shown by the strictly convex properties of the energy functional. Moreover, the orbital stability of every normalized ground state is proven in terms of the Cazenave and Lions' argument.