Band gaps and asymptotic behaviour of continued fraction coefficients

Abstract

Continued fractions are frequently used to approximate densities of states. When the support of these densities is connected (i.e. there is no gap) the coefficients (a n , b n 2 ) of the continued fraction converge when n to infinity . In the presence of gaps, they exhibit undamped oscillations. The authors characterise these oscillations completely by studying the spectrum of periodic linear chains. It is found that the coefficients are periodic abelian functions of n (elliptic functions in the case of a single gap), in agreement with recent results by Magnus (1979). The corresponding recurrence laws are derived, and methods for terminating the continued fraction are given. The case of a single gap is treated in detail, and numerous examples and applications are provided.