Communications in Mathematical Physics

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

Degenerate asymptotic perturbation theory

  • W. Hunziker
  • C. A. Pillet


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.


Neural Network Statistical Physic Complex System Perturbation Theory Nonlinear Dynamics 
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  1. 1.
    Aguilar, J., Combes, J. M.: A class of analytic perturbations for one-body Schrödinger Hamiltonians. Commun. Math. Phys.22, 269–279 (1971)Google Scholar
  2. 2.
    Avron, J. E., Herbest, I. W., Simon, B.: Schrödinger operators with magnetic fields III. Atoms in homogeneous magnetic field. Commun. Math. Phys.79, 529–572 (1981)Google Scholar
  3. 3.
    Baumgärtel, H.: Endlich-dimensionale analytische Störungs-theorie. Berlin: Akademie-Verlag 1972Google Scholar
  4. 4.
    Combes, J. M., Thomas, L.: Asymptotic behaviour of eigenfunctions for multiparticle Schrödinger operators. Commun. Math. Phys.34, 251–270 (1973)Google Scholar
  5. 5.
    Deift, P., Hunziker, W., Simon, B., Vock, E.: Pointwise bounds of eigenfunctions and wave packets inN-Body quantum systems. Commun. Math. Phys.64, 1–34 (1978)Google Scholar
  6. 6.
    Enss, V.: A note on Hunziker's theorem Commun. Math. Phys.52, 233–238 (1977)Google Scholar
  7. 7.
    Herbst, I. W., Simon, B.: Dilation analyticity in constant electric field II.N-Body problem, Borel-summability. Commun. Math. Phys.80, 181–216 (1981)Google Scholar
  8. 8.
    Herbst, I. W.: Temporal exponential decay for the Stark-effect in atoms. Stockholm: Mittag-Leffler Institute, Report No. 16 (1981)Google Scholar
  9. 9.
    Kato, T.: Perturbation theory for linear operators. Berlin, Heidelberg, New York: Springer 1966Google Scholar
  10. 10.
    Reed, M., Simon, B.: Methods of modern mathematical physics. IV. Analysis of operators. New York: Academic Press 1978Google Scholar
  11. 11.
    Vock, E., Hunziker, W.: Stability of Schrödinger eigenvalue problems. Commun. Math. Phys.83, 281–302 (1982)Google Scholar

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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