Rayleigh-Schrödinger series and Birkhoff decomposition

We derive new expressions for the Rayleigh–Schrodinger 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–Schrodinger 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.