Quantum mechanics and semiclassics of hyperbolic n-disk scattering systems

Abstract

The scattering problems of a scalar point particle from an assembly of 1< n<∞ non-overlapping and disconnected hard disks, fixed in the two-dimensional plane, belong to the simplest realizations of classically hyperbolic scattering systems. Their simplicity allows for a detailed study of the quantum mechanics, semiclassics and classics of the scattering. Here, we investigate 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 construct the scattering matrix and its determinant for any non-overlapping n-disk system (with n<∞) and rewrite the determinant in such a way that it separates into the product over 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 multiscattering matrix M which is of Korringa–Kohn–Rostoker-type and which represents the coherent multidisk aspect of the n-disk scattering. Our quantum-mechanical calculation is well-defined at every step, as the on-shell T -matrix and the multiscattering kernel M − 1 are shown to be trace-class. The multiscattering determinant can be organized in terms of the cumulant expansion which is the defining prescription for the determinant over an infinite, but trace-class matrix. The quantum cumulants are then expanded by traces which, in turn, split into quantum itineraries or cycles. These can be organized by a simple symbolic dynamics. The semiclassical reduction of the coherent multiscattering part takes place on the level of the quantum cycles. We show that the semiclassical analog of the mth quantum cumulant is the mth curvature term of the semiclassical zeta function. In this way quantum mechanics naturally imposes the curvature regularization structured by the topological (not the geometrical) length of the pertinent periodic orbits onto the semiclassical zeta function. However, since the cumulant limit m→∞ and the semiclassical limit, ℏ→0 or (wave number) k→∞, do not commute in general, the semiclassical analog of the quantum multiscattering determinant is a curvature expanded semiclassical zeta function which is truncated in the curvature order. We relate the order of this truncation to the topological entropy of the corresponding classical system. We show this explicitly for the three-disk scattering system and discuss the consequences of this truncation for the semiclassical predictions of the scattering resonances. We show that, under the above mentioned truncations in the curvature order, unitarity in n-disk scattering problems is preserved even at the semiclassical level. Finally, with the help of cluster phase shifts, it is shown that the semiclassical zeta function of Gutzwiller and Voros has the correct stability structure and is superior to all the competitor zeta functions studied in the literature.