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arXiv · 2505.06613

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

Yuan-da Wu·Xiaoyu Zeng·Yimin Zhang
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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 $K\nearrow\mathcal{K}_\infty^{(N)}$.

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Yuan-da Wu, Xiaoyu Zeng, Yimin Zhang. 2025-05-10. Gagliardo-Nirenberg-Sobolev inequalities and ground states of Fermions in relativistic Hartree-Fock model. https://arxiv.org/abs/2505.06613

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