Existence of Energy Gaps in the Spectrum of a One-Dimensional Atomic Chain

The question of the existence of an energy gap in the spectrum of a one-dimensional atomic chain is shown to be related to the question of the convergence of a certain continued fraction whose elements are functions of the elements of the two-by-two cell matrices ${\mathbf{C}}^n\mathrm{}\left(E\right)\mathrm{}$ first introduced by Kramers. The implications of a known theorem (Worpitzky's Theorem) in the study of continued fractions are examined for the case of the chain of delta-function potentials and the binary alloy.