Group theoretical approach to band structure and transfer matrices

While dynamical symmetry methods are typically applied to bound and scattering state problems, we consider extensions to periodic potentials. We explore all periodic potentials which can be realized using Lie algebras in two dimensions. We explicitly construct transfer matrices and derive dispersion relations group theoretically. As examples, we show that band structure can be realized in both discrete representations of SU(2) or in projective representations of SU(1,1). For examples in SU(1,1), the band itself belongs to the non-unitary (projective) complementary series. Approaches to band structure as dynamical symmetry problems in higher dimensions are also considered.