Global well-posedness of the nonlinear Hartree equation for infinitely many particles with singular interaction

2024/04/10 by Sonae Hadama, Hadama, Sonae, Younghun Hong +1 · 1 citation
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Quantum chaos and dynamical systems #Quantum many-body systems

paper · pdf · doi:10.48550/arxiv.2404.06730

openalex publication_date 2024/04/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

The nonlinear Hartree equation (NLH) in the Heisenberg picture admits steady states of the form γ_f=f(-Δ) representing quantum states of infinitely many particles. In this article, we consider the time evolution of perturbations from a large class of such steady states via the three-dimensional NLH. We prove that if the interaction potential w has finite measure and initial states have finite relative entropy, then solutions preserve the relative free energy, and they exist globally in time. This result extends the important work of Lewin and Sabin [arXiv:1310.0603] to singular interaction cases.

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• Semi-classical limit of quantum scattering states for the nonlinear Hartree equation

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