  In this letter new, closed and compact analytic expressions for the
evaluation of resonant energies, resonant bound-states, eigenvalues and
eigenfunctions for both scattering and bounded $n$-cell systems are reported.
It is shown that for (scattering and bounded) 1-D systems the eigenfunctions
$\Psi_{\mu ,\nu}(z)$ are simple and well defined functions of the Chebyshev
polynomials of the second kind $U_{n}$, and the energy eigenvalues $E_{\mu ,\nu
}$ (in the $\mu $-th band) are determined by the zeros of these polynomials.
New insights on the energy gap and the localization effect induced by phase
coherence are shown.
