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

Details A direct link between the quantum-mechanical and semiclassical determination of scattering resonances A direct link between the quantum-mechanical and semiclassical determination of scattering resonances

1997-02-06

arxiv.org/abs/chao-dyn/9702004v2

J. Phys. A: Math. Gen. 31 (1998) 2155-2172

Nonlinear Sciences

Chaotic Dynamics

23 pages, latex with IOP journal preprint style, no figures; final version - considerably shortened

Scientific paper

10.1088/0305-4470/31/9/007

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 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 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.

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