From many-body quantum dynamics to the Hartree-Fock and Vlasov equations with singular potentials
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arXiv
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| Natura: | Preprint |
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2021
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| author | Chong, Jacky J. Lafleche, Laurent Saffirio, Chiara |
| author_facet | Chong, Jacky J. Lafleche, Laurent Saffirio, Chiara |
| contents | We obtain the combined mean-field and semiclassical limit from the $N$-body Schrödinger equation for fermions interacting via singular potentials. To obtain the result, we first prove the uniformity in Planck's constant $h$ propagation of regularity for solutions to the Hartree$\unicode{x2013}$Fock equation with singular pair interaction potentials of the form $\pm |x-y|^{-a}$, including the Coulomb and gravitational interactions.
In the context of mixed states, we use these regularity properties to obtain quantitative estimates on the distance between solutions to the Schrödinger equation and solutions to the Hartree$\unicode{x2013}$Fock and Vlasov equations in Schatten norms. For $a\in(0,1/2)$, we obtain local-in-time results when $N^{-1/2} \ll h \leq N^{-1/3}$. In particular, it leads to the derivation of the Vlasov equation with singular potentials. For $a\in[1/2,1]$, our results hold only on a small time scale, or with an $N$-dependent cutoff. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2103_10946 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | From many-body quantum dynamics to the Hartree-Fock and Vlasov equations with singular potentials Chong, Jacky J. Lafleche, Laurent Saffirio, Chiara Analysis of PDEs Mathematical Physics 82C10, 35Q41, 35Q55 (82C05, 35Q83) We obtain the combined mean-field and semiclassical limit from the $N$-body Schrödinger equation for fermions interacting via singular potentials. To obtain the result, we first prove the uniformity in Planck's constant $h$ propagation of regularity for solutions to the Hartree$\unicode{x2013}$Fock equation with singular pair interaction potentials of the form $\pm |x-y|^{-a}$, including the Coulomb and gravitational interactions. In the context of mixed states, we use these regularity properties to obtain quantitative estimates on the distance between solutions to the Schrödinger equation and solutions to the Hartree$\unicode{x2013}$Fock and Vlasov equations in Schatten norms. For $a\in(0,1/2)$, we obtain local-in-time results when $N^{-1/2} \ll h \leq N^{-1/3}$. In particular, it leads to the derivation of the Vlasov equation with singular potentials. For $a\in[1/2,1]$, our results hold only on a small time scale, or with an $N$-dependent cutoff. |
| title | From many-body quantum dynamics to the Hartree-Fock and Vlasov equations with singular potentials |
| topic | Analysis of PDEs Mathematical Physics 82C10, 35Q41, 35Q55 (82C05, 35Q83) |
| url | https://arxiv.org/abs/2103.10946 |