  For a qualitative analysis of spectra of certain two-dimensional
rectangular-well quantum systems several rigorous methods of number theory are
shown productive and useful. These methods (and, in particular, a
generalization of the concept of Markov constant known in Diophantine
approximation theory) are shown to provide a new mathematical insight in the
phenomenologically relevant occurrence of anomalies in the spectra. Our results
may inspire methodical innovations ranging from the description of the
stability properties of metamaterials and of certain hiddenly unitary quantum
evolution models up to the clarification of the mechanisms of occurrence of
ghosts in quantum cosmology.
