Surface states/modes in one-dimensional semi-infinite crystals

Analytical expressions on how the eigenvalue of an existing surface state/mode in one-dimensional semi-infinite periodic systems depends on the boundary location and the boundary condition are obtained, by using a Sturm-Liouville theory approach. The obtained equations are verified by numerical calculations. A direct consequence of the results obtained is that the termination of the periodicity at a boundary τ in a semi-infinite periodic system does not always cause a surface state/mode in a specific band gap.