How good are the dynamical results obtained through different terminators of a continued fraction approach?

Extrapolation procedures for obtaining higher order recurrants within the Method of Recurrence Relation have been analyzed considering one-dimensional spin-1/2 quantum systems comprising the isotropic XY model, the random transverse Ising model, and the isotropic Heisenberg model. The first six recurrants have already been computed for the XY model, as well as nine recurrants for the random transverse Ising model, and fourteen recurrants for the Heisenberg model. Nevertheless, from the exact solution of the autocorrelation function and its spectral density, it has been possible here to exactly compute all orders of the moments and recurrants for the XY model. An analytical expression for the recurrants of the XY model has also been proposed, which is able to furnish all of the recurrants, of any order, to within only 0.03 percent error. An extrapolation process has then been employed to analyze the autocorrelation functions and their spectral densities for all three models. It has been noticed that, as expected, the more recurrants are estimated the longer is the timespan of the computed autocorrelation function. However, the timespan saturates and becomes insensitive as more recurrants are taken into accout. On the other hand, the autocorrelation function needs a high number of recurrants in order to obtain a quantitative account of the spectral function. Even so, the behavior of the spectral density still remains, in cases where just a few recurrants are exactly known, not completely clear in the low frequency region.