On Eigenvalues and Eigenfunctions Absent in the Actual Solid State Theory

Details On Eigenvalues and Eigenfunctions Absent in the Actual Solid State Theory On Eigenvalues and Eigenfunctions Absent in the Actual Solid State Theory

2000-09-05

arxiv.org/abs/cond-mat/0009064v1

Physics

Condensed Matter

Soft Condensed Matter

10 pages, 5 figures

Scientific paper

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.

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