We introduce a formalism, inspired on the perturbation theory for nearly free electrons in a solid-state crystal, to describe the resonances in optical ring resonators subjected to a perturbation in their dielectric profile. We find that, for small perturbations, degenerate resonant modes are split with the splitting proportional to one specific coefficient of the Fourier expansion of the perturbation. We also find an expected asymmetry in the linewidths (and Q factors) of the split modes. Experimental transmission spectra from rings with specially designed perturbations show a qualitative match with the formalism predictions.