Gagliardo-Nirenberg-Sobolev inequalities and ground states of Fermions in relativistic Hartree-Fock model
Fuente:
arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866915282372526080 |
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| author | Wu, Yuan-da Zeng, Xiaoyu Zhang, Yimin |
| author_facet | Wu, Yuan-da Zeng, Xiaoyu Zhang, Yimin |
| contents | 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 $K\nearrow\mathcal{K}_\infty^{(N)}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_06613 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Gagliardo-Nirenberg-Sobolev inequalities and ground states of Fermions in relativistic Hartree-Fock model Wu, Yuan-da Zeng, Xiaoyu Zhang, Yimin Mathematical Physics 35J20, 35J60, 35Q55 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 $K\nearrow\mathcal{K}_\infty^{(N)}$. |
| title | Gagliardo-Nirenberg-Sobolev inequalities and ground states of Fermions in relativistic Hartree-Fock model |
| topic | Mathematical Physics 35J20, 35J60, 35Q55 |
| url | https://arxiv.org/abs/2505.06613 |