On Eigenvalues and Eigenfunctions Absent in the Actual Solid State Theory

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 ${\mathrm{\Psi }}_{\mu ,\nu }(z)$ are simple and well defined functions of the Chebyshev polynomials of the second kind $U_n$, and the energy eigenvalues $E_{\mu ,\nu }$(inthe$\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.