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Communications in Mathematical Physics

, Volume 90, Issue 2, pp 219–233 | Cite as

Degenerate asymptotic perturbation theory

  • W. Hunziker
  • C. A. Pillet
Article

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 ofm×m—matrix functionsA(κ) of a complex variable κ which have an asymptotic expansion εA k κ k as τ→0. We show that asymptotic expansions for groups of eigenvalues and for the corresponding spectral projections ofA(κ) can be obtained from the set {Aκ} by analytic perturbation theory. Special attention is given to the case whereA(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 ofA(κ) which are shown to be holomorphic inS near κ=0 and Borel summable ifA k * =A k for allk. We then fit these results into the scheme of Rayleigh-Schrödinger perturbation theory and we give some examples of asymptotic estimates for Schrödinger operators.

Keywords

Neural Network Statistical Physic Complex System Perturbation Theory Nonlinear Dynamics 
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Copyright information

© Springer-Verlag 1983

Authors and Affiliations

  • W. Hunziker
    • 1
  • C. A. Pillet
    • 1
  1. 1.Institut für Theoretische PhysikETH-HönggerbergZürichSwitzerland

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