QUANTIZATION OF HYPERBOLIC N-SPHERE SCATTERING SYSTEMS IN 3-DIMENSIONS

Most discussions of chaotic scattering systems are devoted to two-dime nsional systems. It is of considerable interest to extend those studie s lo tile. in general, more realistic case of three dimensions, In thi s contest, ii is conceptually important tu investigate tile quality of semi-classical methods as a function of the dimensionality. As il mod el system, we choose various three-dimensional generalizations of the famous three-disk: problem which played a central rule 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 fini te number of nonoverlapping and nontouching hard spheres in three dime nsions. We derive expressions for the scattering matrix S and its dete rminant. The determinant of S decomposes into two parts, tile First on e contains the product of the determinants of the individual one-spher e S-matrices and the second one is given by a ratio involving the dete rminants of a characteristic KKR-type matrix and its conjugate. We jus tify our approach by showing that all formal manipulations in these de rivations are correct and that ail 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 semicla ssical descriptions: The semiclassical limit of the cumulant expansion ui the KKR-type matrix is given by the Gutzwiller-Voros zeta function plus diffractional corrections in the curvature expansion, This conne ction is direct since it is not based on any kind of subtraction schem e involving bounded reference systems. We present numerically computed resonances and compare them with the corresponding data for the simil ar two-dimensional N-disk systems and with semiclassical Calculations. (C) 1997 Academic Press.