Resummation of classical and semiclassical periodic-orbit formulas

The convergence properties of cycle-expanded periodic-orbit expressions for the spectra of classical and semiclassical time evolution operators have been studied for the open three-disk billiard. We present evidence that the semiclassical and perhaps the classical Selberg zeta functions have poles. Applying a Padé approximation on the expansions of the full Euler products, as well as on the individual dynamical zeta functions in the products, we calculate the leading poles and the zeros of the improved expansions with the first few poles removed. The removal of poles tends to change the simple linear exponential convergence of the Selberg zeta functions to an exp{-${\mathit{n}}^2$} decay in the semiclassical case. The classical Selberg zeta function decays like exp{-${\mathit{n}}^{3/2}$}. The leading poles of the jth dynamical zeta function are found to equal the leading zeros of the (j+1)th one: However, in contrast to the zeros, which are all simple, the poles seem without exception to be double. The poles are therefore in general not completely canceled by zeros in the way suggested by Artuso, Aurell, and Cvitanović [Nonlinearity 3, 325 (1990)]. The only complete cancellations of this kind occur in the classical Selberg zeta function between the poles (double) of the first and the zeros (squared) of the second dynamical zeta function. Furthermore, we find strong indications that poles are responsible for the presence of spurious zeros in periodic-orbit quantized spectra and that these spectra can be greatly improved by removing the leading poles, e.g., by using the Padé technique.