A DIRECT LINK BETWEEN THE QUANTUM-MECHANICAL AND SEMICLASSICAL DETERMINATION OF SCATTERING RESONANCES

We investigate the scattering of a point particle from n non-overlappi ng, disconnected hard disks which are fixed in the two-dimensional pla ne and study the connection between the spectral properties of the qua ntum-mechanical scattering matrix and its semiclassical equivalent bas ed on the semiclassical zeta function of Gutzwiller and Voros. We rewr ite the determinant of the scattering matrix in such a way that it sep arates into the product of n determinants of one-disk scattering matri ces-representing the incoherent part of the scattering from the n-disk system-and the ratio of two mutually complex conjugate determinants o f the genuine multi-scattering kernel, M, which is of Korringa-Kohn-Ro stoker-type and represents the coherent multi-disk aspect of the n-dis k scattering. Our result is well defined at every step of the calculat ion, as the on-shell T-matrix and the kernel M-1 are shown to be trace -class. We stress that the cumulant expansion (which defines the deter minant over an infinite, but trace-class matrix) induces the curvature regularization scheme to the Gutzwiller-Voros zeta function and thus leads to a new, well defined and direct derivation of the semiclassica l spectral Function. We show that unitarity is preserved even at the s emiclassical level.