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-Schr\"odinger 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-Schr\"odinger series. Our results provide new combinational formulae and a new way \ff{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.

Mots clés: Perturbation theory; Mould theory; Birkhoff decomposition; Quantum mechanics

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. Letters in Mathematical Physics, 2018, 108 (7), pp.1583-1600. ⟨10.1007/s11005-017-1040-1⟩. ⟨hal-01351108v2⟩