1997/02/28 by Andreas Wirzba, Michael Henseler
Mathematics · Physics and Astronomy · #Mathematical functions and polynomials #Quantum Mechanics and Non-Hermitian Physics #Spectral Theory in Mathematical Physics #chao-dyn #nlin.CD
paper · pdf · doi:10.1088/0305-4470/31/9/007
published as J. Phys. A: Math. Gen. 31 (1998) 2155-2172 · 23 pages, latex with IOP journal preprint style, no figures; final version - considerably shortened
openalex publication_date 1998/03/06 · arxiv created 1998/03/17 · arxiv updated 2009/11/30 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/30
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 one-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, , which is of Korringa-Kohn-Rostoker-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 -matrix and the kernel are shown to be trace-class. We stress that the cumulant expansion (which defines the determinant 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 semiclassical spectral function. We show that unitarity is preserved even at the semiclassical level.