Continued fractions are frequently used to approximate densities of states. When the support
of these densities is connected (ie there is no gap) the coefficients (an, bn 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 …

A very efficient and now widely known method of approximating densities of states is to use
continued fraction expansions (for a review see [1]). As far as the applications to solid state
physics problems are concerned, the method has been first applied to densities of states
with connected supports (ie without gaps), in which case the coefficients of the continued
fraction converge. In the presence of gaps, they exhibit undamped oscillations, whose
characteristics are now well understood. The first detailed study of these oscillations is due …