Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Yavdat Il'yasov, Juntao Sun, Nur Valeev, Shuai Yao

TL;DR

This work addresses the uniqueness of ground state solutions for a defocusing Hartree equation with a nonlocal exchange potential by deploying an inverse optimal problem (IOP) framework. It proves the existence of principal solutions and a corresponding unique minimizer of the IOP, enabling reconstruction of the two-point density $\rho$ from spectral data, along with a dual variational formulation. The results establish a continuous dependence of principal solutions on parameters, provide a robust pathway to obtain ground states (including positivity and stability), and connect primal and dual problems to a coherent inverse-spectral interpretation. The developed IOP methodology offers a systematic tool for nonlocal Schrödinger operators, with potential numerical implementations and extensions to multiparameter inverse problems in quantum systems.

Abstract

We study a generalized defocusing Hartree equation with nonlocal exchange potential and repulsive Hartree--Fock interaction. Using an inverse optimal problem (IOP) approach, we prove the existence and uniqueness of ground state solutions. Additionally, we establish the existence of principal solutions, their continuous dependence on parameters, and a dual variational formulation. The IOP method provides a systematic framework for addressing inverse problems in nonlocal Schrödinger operators and offers new insights into the structure of solutions for defocusing Hartree-type equations.

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

TL;DR

This work addresses the uniqueness of ground state solutions for a defocusing Hartree equation with a nonlocal exchange potential by deploying an inverse optimal problem (IOP) framework. It proves the existence of principal solutions and a corresponding unique minimizer of the IOP, enabling reconstruction of the two-point density from spectral data, along with a dual variational formulation. The results establish a continuous dependence of principal solutions on parameters, provide a robust pathway to obtain ground states (including positivity and stability), and connect primal and dual problems to a coherent inverse-spectral interpretation. The developed IOP methodology offers a systematic tool for nonlocal Schrödinger operators, with potential numerical implementations and extensions to multiparameter inverse problems in quantum systems.

Abstract

We study a generalized defocusing Hartree equation with nonlocal exchange potential and repulsive Hartree--Fock interaction. Using an inverse optimal problem (IOP) approach, we prove the existence and uniqueness of ground state solutions. Additionally, we establish the existence of principal solutions, their continuous dependence on parameters, and a dual variational formulation. The IOP method provides a systematic framework for addressing inverse problems in nonlocal Schrödinger operators and offers new insights into the structure of solutions for defocusing Hartree-type equations.
Paper Structure (7 sections, 16 theorems, 113 equations)

This paper contains 7 sections, 16 theorems, 113 equations.

Table of Contents

  1. Introduction and statement of main results
  2. Preliminaries
  3. Proof of Theorem \ref{['thm1']}
  4. Proof of Theorem \ref{['thm2']}
  5. Proof of Theorem \ref{['thm3']}
  6. Proof of Theorem \ref{['thm4']}
  7. Concluding remarks

Key Result

Theorem 1.1

Assume that $0 < \mu < \min\{N, 4\}$, $\bar{\rho} \in \mathbb{L}_\mu^2$, and $\lambda > \lambda_1(\bar{\rho})$. Then:

Theorems & Definitions (16)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Lemma 2.1: Hardy-Littlewood-Sobolev inequality SimonReedzhang
  • Lemma 2.2
  • Proposition 2.3
  • Proposition 2.4
  • Corollary 2.5
  • Proposition 2.6
  • ...and 6 more