Rayleigh-Schrödinger perturbation theory generalized to eigen-operators in non-commutative rings

A perturbation scheme to find approximate solutions of a generalized spectral problem is presented. The spectral problem is generalized in the sense that the “eigenvalues” searched for, are not real numbers but operators in a non-commutative ring, and the associated “eigenfunctions” do not belong to an Hilbert space but are elements of a module on the non-commutative ring. The method is relevant wherever two sets of degrees of freedom can be distinguished in a quantum system. This is the case for example in rotation-vibration molecular spectroscopy. The article clarifies the relationship between the exact solutions of rotation-vibration molecular Hamiltonians and the solutions of the effective rotational Hamiltonians derived in previous works. It also proposes a less restrictive form for the effective dipole moment than the form considered by spectroscopists so far.