Quantization of Hyperbolic N-Sphere Scattering Systems in Three Dimensions

Abstract

Most discussions of chaotic scattering systems are devoted to two-dimensional systems. It is of considerable interest to extend these studies to the, in general, more realistic case of three dimensions. In this context, it is conceptually important to investigate the quality of semiclassical methods as a function of the dimensionality. As a model system, we choose various three dimensional generalizations of the famous three disk problem which played a central role in the study of chaotic scattering in two dimensions. We present a quantum-mechanical treatment of the hyperbolic scattering of a point particle off a finite number of non-overlapping and non-touching hard spheres in three dimensions. We derive expressions for the scattering matrix S and its determinant. The determinant of S decomposes into two parts, the first one contains the product of the determinants of the individual one-sphere S-matrices and the second one is given by a ratio involving the determinants of a characteristic KKR-type matrix and its conjugate. We justify our approach by showing that all formal manipulations in these derivations are correct and that all the determinants involved which are of infinite dimension exist. Moreover, for all complex wave numbers, we conjecture a direct link between the quantum-mechanical and semiclassical descriptions: The semiclassical limit of the cumulant expansion of the KKR-type matrix is given by the Gutzwiller-Voros zeta function plus diffractional corrections in the curvature expansion. This connection is direct since it is not based on any kind of subtraction scheme involving bounded reference systems. We present numerically computed resonances and compare them with the corresponding data for the similar two-dimensional N-disk systems and with semiclassical calculations.