Geodesic
Semiclassical limit of the Bogoliubov–de Gennes equation
Jacky J. Chong1 orcid ; Laurent Lafleche2 orcid ; Chiara Saffirio3 orcid
1Peking University, Beijing, P. R. China
2École Normale Supérieure de Lyon, Lyon, France
3University of Basel, Basel, Switzerland; University of British Columbia, Vancouver, Canada
EMS surveys in mathematical sciences, Tome 12 (2025) no. 1, pp. 289-321

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DOI

In this paper, we rewrite the time-dependent Bogoliubov–de Gennes (BdG) 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 C2 interaction potentials in terms of a semiclassical optimal transport pseudo-metric. Furthermore, combining our current results with the results of Marcantoni et al. [Ann. Henri Poincaré (2024)], we establish a joint semiclassical and mean-field approximation of the dynamics of a system of spin-21​ Fermions by the Vlasov equation in some weak topology.
DOI : 10.4171/emss/100
Classification : 82C10, 81S30, 35Q55, 35Q83, 82C05
Mots-clés : mean-field limit, semiclassical limit, Bogoliubov–de Gennes equation, many-body Schrödinger equation, Hartree–Fock–Bogoliubov equation, Vlasov equation

Jacky J. Chong  1   ; Laurent Lafleche  2   ; Chiara Saffirio  3

1 Peking University, Beijing, P. R. China
2 École Normale Supérieure de Lyon, Lyon, France
3 University of Basel, Basel, Switzerland; University of British Columbia, Vancouver, Canada 
Jacky J. Chong; Laurent Lafleche; Chiara Saffirio. Semiclassical limit of the Bogoliubov–de Gennes equation. EMS surveys in mathematical sciences, Tome 12 (2025) no. 1, pp. 289-321. doi: 10.4171/emss/100
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