Rayleigh-Schrödinger series and Birkhoff decomposition
arXiv:1608.01110 · doi:10.1007/s11005-017-1040-1
Abstract
We derive new expressions for the Rayleigh-Schrödinger 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-Schrödinger 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.