Rayleigh–Schrödinger series and Birkhoff decomposition

  • Jean-Christophe Novelli
  • Thierry Paul
  • David Sauzin
  • Jean-Yves Thibon


We derive new expressions for the Rayleigh–Schrödinger series describing the perturbation of eigenvalues of quantum Hamiltonians. The method, somehow close to the so-called dimensional renormalization in quantum field theory, involves the Birkhoff decomposition of some Laurent series built up out of explicit fully non-resonant terms present in the usual expression of the Rayleigh–Schrödinger series. Our results provide new combinatorial formulae and a new way of deriving perturbation series in quantum mechanics. More generally we prove that such a decomposition provides solutions of general normal form problems in Lie algebras.


Quantum mechanics Perturbation theory Mould theory Birkhoff decomposition 

Mathematics Subject Classification

34D10 81-08 81Q05 81Q15 



This work has been partially carried out thanks to the support of the A*MIDEX Project (No. ANR-11-IDEX-0001-02) funded by the “Investissements d’Avenir” French Government program, managed by the French National Research Agency (ANR). T.P. thanks also the Dipartimento di Matematica, Sapienza Università di Roma, for its kind hospitality during the elaboration of this work. D.S. thanks Fibonacci Laboratory (CNRS UMI 3483), the Centro Di Ricerca Matematica Ennio De Giorgi and the Scuola Normale Superiore di Pisa for their kind hospitality. This work has received funding from the French National Research Agency under the reference ANR-12-BS01-0017.


  1. 1.
    Born, M.: Vorlesungen über Atommechanik. Springer, Berlin (1925). English translation: The Mechanics of the Atom. Ungar, New York (1927)Google Scholar
  2. 2.
    Connes, A., Kreimer, D.: Renormalization in quantum field theory and the Riemann–Hilbert problem I: the Hopf algebra structure of graphs and the main theorem. Commun. Math. Phys. 210, 249–273 (2000)ADSMathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Degli Esposti, M., Graffi, S., Herczynski, J.: Quantization of the classical Lie algorithm in the Bargmann representation. Ann. Phys. 209(2), 364–392 (1991)ADSMathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Écalle, J.: Les fonctions résurgentes, Publ. Math. d’Orsay (vol. 1: 81-05, vol. 2: 81-06, vol. 3: 85-05) 1981 (1985)Google Scholar
  5. 5.
    Écalle, J.: Six lectures on transseries, analysable functions and the constructive proof of Dulac’s conjecture. In: Schlomiuk, D. (ed) Bifurcations and Periodic Orbits of Vector Fields (Montreal, PQ, 1992), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 408. Kluwer Acad. Publ., Dordrecht, pp. 75–184 (1993)Google Scholar
  6. 6.
    Écalle, J., Vallet, B.: Prenormalization, Correction, and Linearization of Resonant Vector Fields or Diffeomorphisms. Prepub. Orsay, pp. 95–32 (1995)Google Scholar
  7. 7.
    Ebrahimi-Fard, K., Guo, L., Manchon, D.: Birkhoff type decompositions and the Baker–Campbell–Hausdorff recursion. Commun. Math. Phys. 267(3), 821–845 (2006)ADSMathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Heisenberg, W.: Matrix mechanik. Zeitscrift für Physik 33, 879–893 (1925)ADSCrossRefGoogle Scholar
  9. 9.
    Kato, T.: Perturbation theory of linear operators. Springer, Berlin (1988)Google Scholar
  10. 10.
    Manchon, D.: Bogota lectures on Hopf algebras, from basics to applications to renormalization. Comptes-rendus des Rencontres mathématiques de Glanon 2001 (2003)Google Scholar
  11. 11.
    Menous, F.: Formal differential equations and renormalization In: Connes, A., Fauvet, F., Ramis, J.-P. (eds.) Renormalization and Galois Theories. IRMA Lect. Math. Theor. Phys. 15, 229–246 (2009)Google Scholar
  12. 12.
    Menous, F.: From dynamical systems to renormalization. JMP 54, 092702 (2013)ADSMathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Menous, F.: Talk at Paths from, to and in renormalization, Potsdam (2016)Google Scholar
  14. 14.
    Paul, T., Sauzin, D.: Normalization in Lie algebras via mould calculus and applications. Regul. Chaotic Dyn. 22(6), 616–649 (2017)ADSMathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Paul, T., Sauzin, D.: Normalization in Banach scale of Lie algebras via mould calculus and applications. Discret. Contin. Dyn. Syst. A 37, 4461–4487 (2017)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Reed, M., Simon, B.: Methods of modern mathematical physics. I and III. Academic Press Inc., New York (1980)MATHGoogle Scholar
  17. 17.
    Sauzin, D.: Mould expansions for the saddle-node and resurgence monomials. In: Connes, A., Fauvet, F., Ramis, J.-P. (eds.) Renormalization and Galois Theories. IRMA Lectures in Mathematics and Theoretical Physics, vol. 15, pp. 83–163 (2009)Google Scholar
  18. 18.
    von Waldenfels, W.: Zur Charakterisierung Liescher Elemente in freien Algebren. Arch. Math. (Basel) 17, 44–48 (1966)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  • Jean-Christophe Novelli
    • 1
  • Thierry Paul
    • 2
  • David Sauzin
    • 3
  • Jean-Yves Thibon
    • 1
  1. 1.Laboratoire d’Informatique Gaspard-Monge (CNRS - UMR 8049)Université Paris-Est Marne-la-ValléeMarne-la-Vallée Cedex 2France
  2. 2.CMLS, Ecole Polytechnique, CNRSUniversité Paris-SaclayPalaiseau CedexFrance
  3. 3.CNRS UMR 8028 – IMCCEObservatoire de ParisParisFrance

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