Rayleigh-Schrödinger series and Birkhoff decomposition

Jean-Christophe Novelli (1) , Thierry Paul (2) , David Sauzin (3) , Jean-Yves Thibon (1)

1 LIGM - Laboratoire d'Informatique Gaspard-Monge
2 CMLS - Centre de Mathématiques Laurent Schwartz
3 IMCCE - Institut de Mécanique Céleste et de Calcul des Ephémérides

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. More generally we prove that such a decomposition provides solutions of a universal " mould equation ", introduced by two of us in an earlier article, which solves general normal form problems in Lie algebras.

Domaines: Equations aux dérivées partielles [math.AP]; Combinatoire [math.CO]; Systèmes dynamiques [math.DS]; Physique mathématique [math-ph]; Géométrie métrique [math.MG]; Physique Quantique [quant-ph]

Jean-Christophe Novelli, Thierry Paul, David Sauzin, Jean-Yves Thibon. Rayleigh-Schrödinger series and Birkhoff decomposition. 2016. ⟨hal-01351108v1⟩