Résonances près de seuils d'opérateurs magnétiques de Pauli et de Dirac

Sambou, Diomba

Canadian journal of mathematics, Tome 65 (2013) no. 5, pp. 1095-1124

Résumé

Nous considérons les perturbations $H\,:=\,{{H}_{0}}\,+\,V$ et $D\,:=\,{{D}_{0}}\,+\,V$ des Hamiltoniens libres ${{H}_{0}}$ de Pauli et ${{D}_{0}}$ de Dirac en dimension 3 avec champ magnétique non constant, $V$ étant un potentiel électrique qui décroıt super-exponentiellement dans la direction du champ magnétique. Nous montrons que dans des espaces de Banach appropriés, les résolvantes de $H$ et $D$ définies sur le demi-plan supérieur admettent des prolongements méromorphes. Nous définissons les résonances de $H$ et $D$ comme étant les pôles de ces extensions méromorphes. D’une part, nous étudions la répartition des résonances de $H$ prés de l’origine 0 et d’autre part, celle des résonances de $D$ près de $\pm m$ où m est la masse d’une particule. Dans les deux cas, nous obtenons d’abord des majorations du nombre de résonances dans de petits domaines au voisinage de 0 et $\pm m$ . Sous des hypothèses supplémentaires, nous obtenons des développements asymptotiques du nombre de résonances qui entraınent leur accumulation près des seuils 0 et $\pm m$ . En particulier, pour une perturbation $V$ de signe défini, nous obtenons des informations sur la répartition des valeurs propres de $H$ et $D$ près de 0 et $\pm m$ respectivement.

Détail

DOI : 10.4153/CJM-2012-057-7

Mots-clés : 35B34, 35P25, opérateurs magnétiques de Pauli et de Dirac, résonances.

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