Uniqueness of Ground State Solutions for a Defocusing Hartree Equation via Inverse Optimal Problems

Technical Summary: Uniqueness of Ground State Solutions for a Defocusing Hartree Equation via Inverse Optimal Problems

Problem Statement
The paper investigates a generalized defocusing Hartree equation in $\mathbb{R}^N$ ($N \ge 1$) featuring nonlocal exchange potentials and repulsive Hartree–Fock interactions. The equation is given by:
$$-\Delta u + V(x)u - S[\rho]u + \gamma(|x|^{-\mu} * |u|^2)u = \lambda u,$$
where $V(x)$ is an external potential growing at infinity, $\gamma > 0$ represents the defocusing (repulsive) interaction, $0 \le \mu < \min\{N, 4\}$, and $S[\rho]$ is a nonlocal exchange operator defined by a symmetric kernel $\rho \in L^2_\mu(\mathbb{R}^N \times \mathbb{R}^N)$.

The primary mathematical challenges addressed are:

1. Existence and Uniqueness: Proving the existence and uniqueness of ground state solutions and principal solutions for this equation.
2. Methodological Limitations: Classical techniques for proving uniqueness, such as symmetric decreasing rearrangement and the moving plane method, are inapplicable due to the presence of the exchange potential $S[\rho]$ and the nonlocal defocusing term. Furthermore, the associated energy functional lacks convexity and phase invariance.
3. Inverse Spectral Context: The study aims to connect the solution of this nonlinear PDE to inverse spectral problems, specifically reconstructing the interaction kernel from spectral data.

Methodology: Inverse Optimal Problems (IOP)
The authors employ a specialized approach based on Inverse Optimal Problems (IOP), a framework previously developed by Ilyasov and Valeev for Sturm-Liouville operators but extended here to nonlocal Schrödinger operators.

The core of the methodology involves formulating a constrained optimization problem (Problem P):
Given a reference kernel $\bar{\rho} \in L^2_\mu$ and a target eigenvalue $\lambda$, find a minimizer $\hat{\rho} \in L^2_\mu$ that minimizes the distance $\|\bar{\rho} - \rho\|_{L^2_\mu}$ subject to the constraint that $\lambda$ is the principal eigenvalue of the operator $L[\rho] = -\Delta + V - S[\rho]$.

Key analytical tools used include:

• Variational Characterization: The principal eigenvalue $\lambda_1(\rho)$ is characterized via a Rayleigh quotient.
• Concavity and Semicontinuity: The functional $\lambda_1(\rho)$ is shown to be concave and upper semicontinuous on $L^2_\mu$. This ensures the existence of minimizers for the IOP and allows for the application of the Kuhn-Tucker theorem.
• Gateaux Differentiability: The differentiability of the principal eigenvalue functional with respect to the kernel $\rho$ is established, linking the variation of the eigenvalue to the eigenfunction.
• Dual Formulation: A dual inverse optimal problem is defined to characterize the relationship between the spectral gap and the norm of the solution.

Key Contributions and Results

1. Existence and Uniqueness of Principal Solutions (Theorem 1.1):
   For $\lambda > \lambda_1(\bar{\rho})$, the paper proves the existence of a unique minimizer $\hat{\rho}$ for the IOP. Crucially, the principal solution $\hat{u}$ of the Hartree equation is recovered pointwise almost everywhere from this minimizer via the relation:
   $$\hat{\rho}(x, y) = \bar{\rho}(x, y) - \hat{u}(x)\hat{u}(y).$$
   This establishes a direct link between the solution of the nonlinear PDE and the solution of the inverse optimization problem.

2. Uniqueness of Ground States (Theorem 1.2):
   Under the condition that the reference kernel $\bar{\rho} \ge 0$, the paper proves the existence of a stable ground state. Furthermore, it establishes that this ground state is unique and positive. The principal solution coincides with this unique ground state.

   • Significance: This result is notable because it provides a uniqueness proof for a defocusing Hartree equation with nonlocal interactions, a class of equations where standard uniqueness methods fail. The paper explicitly notes that this uniqueness result holds even in the case where the exchange potential vanishes ($\rho \equiv 0$), a scenario not previously established in the literature for this specific equation.

3. Continuous Dependence (Theorem 1.3):
   The authors demonstrate that the set of principal solutions forms a continuous branch. Specifically, the map from the kernel $\rho$ to the solution $\hat{u}$ is continuous in the $W$-norm, and the map from the spectral parameter $\lambda$ to the solution is also continuous.

4. Dual Variational Formulation (Theorem 1.4):
   The paper introduces a dual problem involving the maximization of the principal eigenvalue subject to a fixed distance constraint $\|\bar{\rho} - \rho\| = \kappa$. It proves the existence of a unique maximizer and shows that the corresponding principal solution can be recovered via the dual minimizer.

Significance and Claims
The paper claims to offer a systematic framework for addressing inverse problems in nonlocal Schrödinger operators. Its significance lies in three main areas:

• Methodological Innovation: It successfully adapts the Inverse Optimal Problem approach to nonlocal operators, overcoming the limitations of classical rearrangement and moving plane methods.
• Structural Insight: It provides explicit reconstruction formulas for solutions ($\hat{u}$) and kernels ($\hat{\rho}$) in terms of spectral data and reference functions, offering a new perspective on the structure of defocusing Hartree-type equations.
• Theoretical Extension: The work extends the theory of inverse spectral problems beyond Sturm-Liouville operators to a wider class of nonlinear Schrödinger equations with nonlocal interactions.

The authors conclude that this framework not only solves the specific nonlinear equation but also lays the groundwork for generalizing these techniques to multiparameter inverse problems (reconstructing densities from finite sets of eigenvalues) and applications in multi-component quantum systems. They suggest that the variational nature of the problem makes it amenable to numerical approximation, including modern machine learning approaches like the Deep Ritz method.