This talk presents a method for the numerical investigation of the distribution of the
eigenvalues introduced into a spectral gap of a periodic Dirac system by a perturbation
of the type of the angular momentum term.
A number of examples illustrate the effectiveness of the method and demonstrate the remarkable
accuracy of the strong coupling asymptotic formula even for small values of the perturbation
These results also shed some light on the spectrum in the exceptional gap of
radially periodic three-dimensional Dirac operators.