  We show that dynamical symmetry methods can be applied to Hamiltonians with
periodic potentials. We construct dynamical symmetry Hamiltonians for the Scarf
potential and its extensions using representations of su(1,1) and so(2,2).
Energy bands and gaps are readily understood in terms of representation theory.
We compute the transfer matrices and dispersion relations for these systems,
and find that the complementary series plays a central role as well as
non-unitary representations.
