In this talk I will describe how to compute sharp intervals of enclosure for the eigenvalues of matrix differential operators with singular coefficients. My main concern will be the angular Kerr–Newman Dirac operator with the possible addition of a smooth perturbation. For the unperturbed model I will show how to validate existing benchmarks as well as how to sharpen them by several orders of magnitude. As a main tool I will employ a pollution-free Galerkin-type method known as the quadratic method. The research is based on recent joint work with Monika Winklmeier.