Quantum systems, spectrum generating algebras and non-compact lie algebras

Spectrum generating algebras are examined for non-compact Lie algebras where projective representations can accommodate Hamiltonians with periodic potentials and Bloch wavefunctions. It is here that the representations such as the projective complementary series of S U(1, 1) finds application in the calculation of dispersion relations, band structure and transfer matrices. We comment on the classes of representations, the need for unitarity and other issues that arise in such systems. Limiting forms and coordinate transformations allow the derivation of many classes of periodic and non-periodic potentials, including bound, scattering and band structure states. These are all mapped back to the classes of projective representations that form the basis for the physical states.