Application of group ring algebra to localized and delocalized quantum states in periodic potentials

The quantum system having cosinusoidal potential energy is a well known model in which the Schrödinger equation can be cast in the form of the Mathieu equation. The periodic eigenfunctions of the Mathieu equation are then related to the wavefunctions associated with a Hamiltonian having a cosinusoidal potential energy term. These wavefunctions are well known but they are delocalized functions having amplitude in more than one potential energy well. Sometimes a more convenient set of functions are those with amplitude localized in only one of the potential energy wells. This system is analyzed in the context of the group ring algebra associated with the symmetry of the Hamiltonian. Primary results of this work are (i) abstraction to the group ring algebra associated with these systems and (ii) the presentation of general formulae for the localized wavefunctions in terms of linear combinations of the delocalized wavefunctions. Explicit analytic results are presented for cases where the cosinusoidal potential energy has two through nine wells. Discussion of how the group ring approach serves as a path to other types of periodic potentials is also given.