Divergences of the semiclassical S-matrix formula in irregular scattering

The convergence condition of the Millerʹs semiclassical S-matrix formula in the case of irregular inelastic scattering is discussed. The discussion does not refer directly to properties of dynamics, which allows one to obtain results for smooth scattering systems with mixed phase space that are by far more difficult for analysis than hyperbolic systems. The absolute convergence of the semiclassical S-matrix is determined by the global geometrical properties of fractal iciclic structures appearing in the excitation profile of irregular scattering. It is shown that such properties can be characterized by a single parameter that we call convergence dimension DC; the convergence criterion reads DC < 12. We present the conjecture that DC for Cantor-like sets is identical with the Hausdorff dimension DH. This conjecture is shown to be true for several representative models of fractal iciclic structure, including a dynamics-related model.