Hydrogenic systems confined by infinite tubes

We consider a hydrogen atom or ion confined within a straight infinite tube. Modelled by an appropriate nonrelativistic Schrödinger operator with Dirichlet boundary conditions, basic properties of the discrete and continuous spectrum of the system are investigated by rigorous methods. This includes the behaviour of energies and ionization potentials when the size of the confining region is changing. For the particular case of cylindrical tubes, classical symmetries are analysed in terms of geometrical mechanics and the results employed for a systematic construction of corresponding symmetry operators in the quantum regime.