Letters in Mathematical Physics

, Volume 108, Issue 7, pp 1583–1600 | Cite as

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.


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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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