Group Theory Approach to Band Structure: Scarf and Lame Hamiltonians

TL;DR

This paper extends group theoretical methods to analyze band structures in quantum systems, deriving transfer matrices and dispersion relations using symmetries from compact and non-compact groups.

Contribution

It introduces a novel group theoretical framework for band structure analysis, incorporating dynamical symmetries and algebraic methods for Hamiltonians with periodic potentials.

Findings

Derived transfer matrices for band structures

Established dispersion relations from symmetry considerations

Identified roles of compact and non-compact groups

Abstract

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.