  The group theoretical treatment of bound and scattering state problems is
extended to include band structure. We show that one can realize Hamiltonians
with periodic potentials as dynamical symmetries, where representation theory
provides analytic solutions, or which can be treated with more general spectrum
generating algebraic methods. We find dynamical symmetries for which we derive
the transfer matrices and dispersion relations. Both compact and non-compact
groups are found to play a role.
