Rayleigh-Schr\"odinger series and Birkhoff decomposition

TL;DR

This paper introduces new combinational formulas for the Rayleigh-Schr"odinger perturbation series in quantum mechanics, utilizing Birkhoff decomposition of Laurent series, and extends the approach to normal form problems in Lie algebras.

Contribution

It presents a novel method combining Birkhoff decomposition with perturbation series derivation, offering a new perspective and tools for quantum perturbation theory and Lie algebra normal forms.

Findings

New explicit formulas for Rayleigh-Schr"odinger series

Application of Birkhoff decomposition to quantum perturbations

Solutions to normal form problems in Lie algebras

Abstract

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.