This Note describes various analytical and computational results concerning the calculation of Dirac eigenvalues, or more generally, of operators with gaps. An algorithm based on an abstract theorem characterizing the eigenvalues in gaps was found years ago, but it is only recently that a delicate analysis to identify and study the domains of those operators has allowed to put that algorithm on a firm basis concerning the choice of approximation basis sets, and this both for light and for heavy atoms. The works described here concern joint papers with several collaborators: J. Dolbeault, M. Lewin, M. Loss, E. Séré and M. Vanbreugel.
Cette Note présente divers résultats analytiques et numériques concernant le calcul des valeurs propres de l’opérateur de Dirac, ou plus généralement, des opérateurs avec des “gaps” spectraux. Un algorithme basé sur un théorème abstrait caractérisant les valeurs propres dans les écarts a été trouvé il y a des années, mais ce n’est que récemment qu’une analyse délicate pour identifier et étudier les domaines de ces opérateurs a permis de mettre cet algorithme sur une base ferme concernant le choix des approximations, et ceci aussi bien pour les atomes légers que pour les atomes lourds. Les travaux décrits ici concernent des travaux réalisés avec plusieurs collaborateurs : J. Dolbeault, M. Lewin, M. Loss, E. Séré et M. Vanbreugel.
Mots-clés : Opérateur de Dirac, Valeur propre, Domaine de l’opérateur, Calcul des valeurs propres, Opérateur auto-adjoint
Maria J. Esteban 1
CC-BY 4.0
@article{CRPHYS_2020__21_2_177_0,
author = {Maria J. Esteban},
title = {Mathematical questions about the computation of eigenvalues of {Dirac} operators with critical potentials in atomic and molecular physics},
journal = {Comptes Rendus. Physique},
pages = {177--183},
publisher = {Acad\'emie des sciences, Paris},
volume = {21},
number = {2},
year = {2020},
doi = {10.5802/crphys.16},
language = {en},
}
TY - JOUR AU - Maria J. Esteban TI - Mathematical questions about the computation of eigenvalues of Dirac operators with critical potentials in atomic and molecular physics JO - Comptes Rendus. Physique PY - 2020 SP - 177 EP - 183 VL - 21 IS - 2 PB - Académie des sciences, Paris DO - 10.5802/crphys.16 LA - en ID - CRPHYS_2020__21_2_177_0 ER -
%0 Journal Article %A Maria J. Esteban %T Mathematical questions about the computation of eigenvalues of Dirac operators with critical potentials in atomic and molecular physics %J Comptes Rendus. Physique %D 2020 %P 177-183 %V 21 %N 2 %I Académie des sciences, Paris %R 10.5802/crphys.16 %G en %F CRPHYS_2020__21_2_177_0
Maria J. Esteban. Mathematical questions about the computation of eigenvalues of Dirac operators with critical potentials in atomic and molecular physics. Comptes Rendus. Physique, Prizes of the French Academy of Sciences 2019, Volume 21 (2020) no. 2, pp. 177-183. doi : 10.5802/crphys.16. https://comptes-rendus.academie-sciences.fr/physique/articles/10.5802/crphys.16/
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