Degenerate asymptotic perturbation theory

Abstract

Asymptotic Rayleigh-Schrödinger perturbation theory for discrete eigenvalues is developed systematically in the general degenerate case. For this purpose we study the spectral properties of m×m-matrix functions A(κ) of a complex variable κ which have an asymptotic expansion εAkκk as τ→0. We show that asymptotic expansions for groups of eigenvalues and for the corresponding spectral projections of A(κ) can be obtained from the set {Aκ} by analytic perturbation theory. Special attention is given to the case where A(κ) is Borel-summable in some sector originating from κ=0 with opening angle >π. Here we prove that the asymptotic series describe individual eigenvalues and eigenprojections of A(κ) which are shown to be holomorphic in S near κ=0 and Borel summable if Ak*=Ak for all k. We then fit these results into the scheme of Rayleigh-Schrödinger perturbation theory and we give some examples of asymptotic estimates for Schrödinger operators. © 1983 Springer-Verlag.