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

TL;DR

This paper rigorously analyzes the relativistic Hartree-Fock model for finite Fermi systems, establishing key inequalities, characterizing ground states, and exploring their asymptotic behavior near critical coupling thresholds.

Contribution

It introduces an optimal Gagliardo-Nirenberg-Sobolev inequality with Hartree nonlinearities and characterizes ground state existence in relation to this inequality's constant.

Findings

Established an optimal GNS inequality for orthonormal systems.

Derived a finite-rank Lieb-Thirring inequality dual to the GNS inequality.

Proved ground states exist iff the coupling parameter is below a critical value.

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 $K<\mathcal{K}_\infty^{(N)}$, where $\mathcal{K}_\infty^{(N)}$ 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 $K\nearrow\mathcal{K}_\infty^{(N)}$.