[Calcul de l'indice de Maslov et du flux spectral au moyen de signatures partielles]
Given a smooth Lagrangian path, both in the finite and in the infinite dimensional (Fredholm) case, we introduce the notion of partial signatures at each isolated intersection of the path with the Maslov cycle. For real-analytic paths, we give a formula for the computation of the Maslov index using the partial signatures; a similar formula holds for the spectral flow of real-analytic paths of Fredholm self-adjoint operators on real separable Hilbert spaces. As applications of the theory, we obtain a semi-Riemannian version of the Morse index theorem for geodesics with possibly conjugate endpoints, and we prove a bifurcation result at conjugate points along semi-Riemannian geodesics.
Etant donné un chemin régulier de lagrangiens, nous introduisons dans le cas de la dimension finie et le cas (Fredholm) de dimension infinie la notion de signatures partielles en chaque intersection isolée d'un tel chemin avec le cycle de Maslov. En utilisant les signatures partielles, nous donnerons une formule de calcul de l'indice de Maslov. Une formule semblabe vaut pour le flux spectral de chemins réel-analytiques d'opérateurs auto-adjoints de Fredholm sur des espaces de Hilbert réels et séparables. Comme application de la théorie, nous obtenons une version semi-Riemannienne du théorème de l'indice de Morse dans le cas de géodésiques avec des points initiaux conjugués. Enfin, nous démontrons un résultat de bifurcation en ces points conjugués le long des géodésiques semi-Riemanniennes.
Accepté le :
Publié le :
Roberto Giambò 1 ; Paolo Piccione 2 ; Alessandro Portaluri 2
@article{CRMATH_2004__338_5_397_0, author = {Roberto Giamb\`o and Paolo Piccione and Alessandro Portaluri}, title = {Computation of the {Maslov} index and the spectral flow via partial signatures}, journal = {Comptes Rendus. Math\'ematique}, pages = {397--402}, publisher = {Elsevier}, volume = {338}, number = {5}, year = {2004}, doi = {10.1016/j.crma.2004.01.004}, language = {en}, }
TY - JOUR AU - Roberto Giambò AU - Paolo Piccione AU - Alessandro Portaluri TI - Computation of the Maslov index and the spectral flow via partial signatures JO - Comptes Rendus. Mathématique PY - 2004 SP - 397 EP - 402 VL - 338 IS - 5 PB - Elsevier DO - 10.1016/j.crma.2004.01.004 LA - en ID - CRMATH_2004__338_5_397_0 ER -
Roberto Giambò; Paolo Piccione; Alessandro Portaluri. Computation of the Maslov index and the spectral flow via partial signatures. Comptes Rendus. Mathématique, Volume 338 (2004) no. 5, pp. 397-402. doi : 10.1016/j.crma.2004.01.004. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2004.01.004/
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