Convolution of the secular equation of perturbation theory for a degenerate case

A simple procedure of the convolution of secular equation of perturbation theory in the degenerate case is suggested which is valid for the arbitrary multiplicity of the unperturbed level degeneration. The expressions obtained can be considered as an extension of the Brilliouin–Wigner perturbation theory to the case of degenerate of almost degenerate states. Some equivalent formulations of the Schrödinger equation are obtained which might be useful in concrete calculations. Problem of an electron in the periodic harmonic potential is considered as an example.