Exact confined eigenstates in one-dimensional finite periodic systems with arbitrary symmetric basis

A set of eigenenergy equations for complete confinement of N periodic one-dimensional system with arbitrary symmetric potential in each cell is presented. Exact eigenenergies and their eigenfunctions are solved analytically. We find different solutions on the band edge states comparing early results solved by conventional methods. Our results show that higher band edge state of each bandgap is dependent on the confined length, but lower band edge state is independent of the length. Moreover, wavefunctions for lower band edge states are Bloch waves but those for other eigenstates, including higher band edge states, do not obey Bloch waves.