Dynamical symmetries of the perturbed hydrogen atom: The van der Waals interaction

We have studied theoretically the spectrum of hydrogen atoms subjected to a certain class of perturbations, including the van der Waals and the diamagnetic interactions. We find a universal adiabatic invariant for this class. In special cases the system exhibits dynamical symmetry and an analytic spectrum. The pattern of energy levels in general can be understood in terms of the adjacent symmetries.