Degenerate asymptotic perturbation theory
Résumé
Asymptotic Rayleigh-Schrodinger perturbation theory for discrete eigenvalues is developed systematically in the general degenerate case. For this purpose we study the spectral properties of mxm-matrix functions A(kappa) of a complex variable kappa which have an asymptotic expansion $\sum_j A_j \kappa^j$ as kappa->0. We show that asymptotic expansions for groups of eigenvalues and for the corresponding spectral projections of A(kappa) can be obtained from the set $\{A_j\}$ by analytic perturbation theory. Special attention is given to the case where A(kappa) is Borel-summable in some sector originating from kappa=0 with opening angle larger than pi. Here we prove that the asymptotic series describe individual eigenvalues and eigenprojections of A(kappa) which are shown to be holomorphic in S near kappa=0 and Borel summable if $A_j\ast=A_j$ for all j. We then fit these results into the scheme of Rayleigh-Schrodinger perturbation theory and we give some examples of asymptotic estimates for Schrdinger operators.