  Long-period systems and superlattices, with additional periodicity, have new
effects on the energy spectrum and wave functions. Most approaches adjust
theories for infinite systems, which is acceptable for large but not small
number of unit cells $n$. In the past 30 years, a theory based entirely on
transfer matrices was developed, where the finiteness of $n$ is an essential
condition. The theory of finite periodic systems (TFPS) is also valid for any
number of propagating modes, and arbitrary potential profiles (or refractive
indices). We review this theory, the transfer matrix definition, symmetry
properties, group representations, and relations with the scattering
amplitudes. We summarize the derivation of multichannel matrix polynomials
(which reduce to Chebyshev polynomials in the one-propagating mode limit), the
analytical formulas for resonant states, energy eigenvalues, eigenfunctions,
parity symmetries, and discrete dispersion relations, for superlattices with
different confinement characteristics. After showing the inconsistencies and
limitations of hybrid approaches that combine the transfer-matrix method with
Floquet's theorem, we review some applications of the TFPS to multichannel
negative resistance, ballistic transistors, channel coupling, spintronics,
superluminal, and optical antimatter effects. We review two high-resolution
experiments using superlattices: tunneling time in photonic band-gap and
optical response of blue-emitting diodes, and show extremely accurate
theoretical predictions.
