Gagliardo-Nirenberg-Sobolev inequalities and ground states of Fermions in relativistic Hartree-Fock model

2025/05/10 by Yuanda Wu, Wu, Yuan-da, Xiaoyu Zeng +3
Mathematics · #35J20 #35J60 #35Q55 #Advanced Mathematical Physics Problems #FOS: Physical sciences #Mathematical Physics (math-ph) #Nonlinear Partial Differential Equations #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2505.06613

openalex publication_date 2025/05/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This paper presents a rigorous mathematical analysis of the relativistic Hartree-Fock model for finite Fermi systems. We first establish an optimal Gagliardo-Nirenberg-Sobolev (GNS) inequality with Hartree-type nonlinearities for orthonormal systems and characterize the qualitative properties of its optimizers. Furthermore, we derive a finite-rank Lieb-Thirring inequality involving convolution terms and show that it is the duality of the GNS-inequality-a result that, to our knowledge, has not previously appeared in the literature. For the relativistic Hartree-Fock model, we prove that ground states exist if and only if the coupling parameter K<K_∞^(N), where K_∞^(N) is the optimal constant in the GNS-inequality. Finally, under suitable assumptions on the external potentials, we calculate the precisely asymptotic behavior of ground states as K\nearrowK_∞^(N).

Citations

• Mass-Critical Neutron Stars in the Hartree-Fock and Hartree-Fock-Bogoliubov Theories
• Mass Concentration of Two-Spinless Fermi Systems with Attractive Interactions
• Ground States of Fermionic Nonlinear Schrödinger Systems with Coulomb Potential II: The L²-Critical Case
• Ground States of Fermionic Nonlinear Schrödinger Systems with Coulomb Potential I: The L²-Subcritical Case
• Lieb-Thirring constant on the sphere and on the torus
• Properties of ground states of attractive Gross-Pitaevskii equations with multi-well potentials
• Local and global minimizers for a variational energy involving a\n fractional norm
• On ground states for the L²-critical boson star equation
• Lieb-Thirring inequalities with improved constants
• Mean Field Dynamics of Boson Stars
• Sharp Lieb-Thirring Inequalities in High Dimensions

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• Hartree-Fock type systems: existence of ground states and asymptotic behavior
• Existence and symmetry breaking of vectorial ground states for Hartree-Fock type systems with potentials
• On global classical solutions of the time-dependent von Neumann equation for Hartree-Fock systems
• Existence of Minimizers in Restricted Hartree-Fock Theory