The link between the quantum-mechanical and semiclassical determination of scattering resonances

We investigate the scattering of a point particle from n non-overlapping, disconnected hard disks which are fixed in the two-dimensional plane and study the connection between the spectral properties of the quantum-mechanical scattering matrix and its semiclassical equivalent based on the semiclassical zeta-function of Gutzwiller and Voros. We rewrite the determinant of the scattering matrix in such a way that it separates into the product of n determinants of 1-disk scattering matrices – representing the incoherent part of the scattering from the n disk system – and the ratio of two mutually complex conjugate determinants of the genuine multi-scattering kernel, M, which is of KKR-type and represents the coherent multi-disk aspect of the n-disk scattering. Our result is well-defined at every step of the calculation, 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 determinant over an infinite, but trace class matrix) imposes the curvature regularization scheme to the Gutzwiller-Voros zeta function and thus leads to a new, well-defined and direct derivation of the semiclassical spectral function. We show that unitarity is preserved even at the semiclassical level. We discuss the convergence properties of cumulant and curvature expansions. PACS numbers: 03.65.Sq, 03.20.+i, 05.45.+b Short title: The link between quantum-mechanical and semiclassical scattering