, De surcroît, le cas de certains potentiels dégénérés, i.e., lim inf |x|?? V (x) = ??, peut être considéré. Si l'on revient sur l'exemple typique de la loi de Subbotin de paramètre ? > 1 (on rappelle que pour ? > 1 le spectre est discret tandis que pour ? = 1 et la loi de Laplace, le trou spectral existe et vaut 1/4, mais correspond exactement au bas du spectre essentiel de l'opérateur), on obtient le résultat suivant. Comme pour le trou spectral, nous nous concentrons plutôt sur la borne inférieure car c, Le Théorème 3.2.24 peut être appliqué dans de nombreuses situations pour lesquelles le potentiel V n'est pas uniformément convexe, voire même convexe

, Dans le cadre de la loi de Subbotin de paramètre ? > 1, on a : ? si ? ?]1, 2[, alors pour tout ? > 0, il existe une constante C ?,? > 0 explicite en fonction des paramètres ? et ? telle que pour tout n ? N *

, ? si ? ? 2, alors il existe C ? > 0 explicite telle que pour tout n ? N *

, Ces résultats sont en accord avec la loi de Weyl donnant le comportement asymptotique de la fonction de comptage des valeurs propres pour tout ? > 1, cf

, Concluons la présentation de l'article [38] en énonçant le dernier résultat que nous souhaiterions évoquer. L'identité (3.2.15) ouvre la voie, comme dans le cas trivial d'un potentiel uniformément convexe, cf. (3.2.12), à une estimation de l'écart entre les valeurs propres, la difficulté résidant dans le caractère non explicite des poids a i car dépendant des fonctions propres g i qui sont en général inconnues. Malgré cette anicroche, l'écart au moins entre les deux premières valeurs propres strictement positives peut être estimé en utilisant

, Notons que cette question n'a d'intérêt que si ? 1 (?L) ? ? disc (?L) (en revanche il se peut que ? 2 (?L) ? ? ess (?L)). Dans le cas présent, la densité de la mesure invariante µ a de l'opérateur ?L a est proportionnelle à (g 1 ) 2 e ?V , c'est-a-dire qu'elle dépend de la fonction propre g 1 inconnue. Néanmoins, si le potentiel associé V a est uniformément convexe, alors on a l'inégalité ? 1 (?L a ) ? inf R V a et le tour est joué. Ainsi, le travail consiste à montrer cette propriété de convexité uniforme sous certaines hypothèse sur V et

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