Semiclassical Limit of the Bogoliubov-de Gennes Equation
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arXiv
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| Formato: | Preprint |
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2024
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| _version_ | 1866913935388573696 |
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| author | Chong, Jacky J. Lafleche, Laurent Saffirio, Chiara |
| author_facet | Chong, Jacky J. Lafleche, Laurent Saffirio, Chiara |
| contents | In this paper, we rewrite the time-dependent Bogoliubov$\unicode{x2013}$de Gennes equation in an appropriate semiclassical form and establish its semiclassical limit to a two-particle kinetic transport equation with an effective mean-field background potential satisfying the one-particle Vlasov equation. Moreover, for some semiclassical regimes, we obtain a higher-order correction to the two-particle kinetic transport equation, capturing a nontrivial two-body interaction effect. The convergence is proven for $C^2$ interaction potentials in terms of a semiclassical optimal transport pseudo-metric. Furthermore, combining our current results with the results of Marcantoni et al. [arXiv:2310.15280], we establish a joint semiclassical and mean-field approximation of the dynamics of a system of spin-$\frac{1}{2}$ Fermions by the Vlasov equation in some negative order Sobolev topology. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_15880 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Semiclassical Limit of the Bogoliubov-de Gennes Equation Chong, Jacky J. Lafleche, Laurent Saffirio, Chiara Mathematical Physics Analysis of PDEs Quantum Physics 82C10 (Primary) 81S30, 35Q55, 35Q83, 82C05 (Secondary) In this paper, we rewrite the time-dependent Bogoliubov$\unicode{x2013}$de Gennes equation in an appropriate semiclassical form and establish its semiclassical limit to a two-particle kinetic transport equation with an effective mean-field background potential satisfying the one-particle Vlasov equation. Moreover, for some semiclassical regimes, we obtain a higher-order correction to the two-particle kinetic transport equation, capturing a nontrivial two-body interaction effect. The convergence is proven for $C^2$ interaction potentials in terms of a semiclassical optimal transport pseudo-metric. Furthermore, combining our current results with the results of Marcantoni et al. [arXiv:2310.15280], we establish a joint semiclassical and mean-field approximation of the dynamics of a system of spin-$\frac{1}{2}$ Fermions by the Vlasov equation in some negative order Sobolev topology. |
| title | Semiclassical Limit of the Bogoliubov-de Gennes Equation |
| topic | Mathematical Physics Analysis of PDEs Quantum Physics 82C10 (Primary) 81S30, 35Q55, 35Q83, 82C05 (Secondary) |
| url | https://arxiv.org/abs/2403.15880 |