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

TL;DR

This work analyzes the Hartree equation with a harmonic potential in $\mathbb{R}^N$ under a prescribed mass constraint. The authors employ a mass-energy constrained variational framework to prove the existence of normalized ground states for every $m>0$ and identify the negative frequency $\omega<0$. They establish uniqueness of normalized ground states by converting to a density variable $\rho=|u|^2$ and exploiting strict convexity of the transformed energy $\widetilde{\mathcal{E}}(\sqrt{\rho})$, yielding ground states up to a phase as $Q e^{i\theta}$. Finally, they prove orbital stability of all normalized ground states via the Cazenave-Lions approach, showing solutions starting near the orbit stay close for all times. These results deepen the understanding of variational structure and dynamics of nonlocal nonlinear Schrödinger equations with confinement.

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.

Theorems & Definitions (11)

• Proposition 2.1
• Proposition 2.2
• Proposition 2.3
• Proposition 2.4
• Theorem 3.1
• Theorem 3.2
• Lemma 4.1
• Lemma 4.2
• Theorem 4.3
• Lemma 5.1