  We derive new expressions for the Rayleigh-Schr\"odinger seriesdescribing the
perturbation of eigenvalues of quantumHamiltonians. The method, somehow close
to the so-called dimensionalrenormalization in quantum field theory, involves
the Birkhoffdecomposition of some Laurent series built up out of explicit
fullynon-resonant terms present in the usual expression of
theRayleigh-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.
