Wirzba, Andreas
:
Quantum Mechanics and Semiclassics of Hyperbolic nDisk Scattering Systems.
TU Darmstadt
[Habilitation], (2008)

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Item Type:  Habilitation  

Title:  Quantum Mechanics and Semiclassics of Hyperbolic nDisk Scattering Systems  
Language:  English  
Abstract:  The scattering problems of a scalar point particle from a finite assembly of n>1 nonoverlapping and disconnected hard disks, fixed in the twodimensional plane, belong to the simplest realizations of classically hyperbolic scattering systems. Here, we investigate the connection between the spectral properties of the quantummechanical scattering matrix and its semiclassical equivalent based on the semiclassical zetafunction of Gutzwiller and Voros. Our quantummechanical calculation is welldefined at every step, as the onshell Tmatrix and the multiscattering kernel M1 are shown to be traceclass. The multiscattering determinant can be organized in terms of the cumulant expansion which is the defining prescription for the determinant over an infinite, but traceclass 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>infinity and the semiclassical limit hbar>0 do not commute in general, the semiclassical analog of the quantum multiscattering determinant is a curvature expanded, truncated semiclassical zeta function. We relate the order of this truncation to the topological entropy of the corresponding classical system. 

Uncontrolled Keywords:  quantph  
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Classification DDC:  500 Naturwissenschaften und Mathematik > 530 Physik  
Divisions:  Fachbereich Physik  
Date Deposited:  17 Oct 2008 09:23  
Last Modified:  07 Dec 2012 11:54  
URN:  urn:nbn:de:tudatuprints11301  
Additional Information:  PACS96:03.65.Sq Semiclassical theories and applications PACS96:03.20.+i Classical mechanics of discrete systems: general mathematical aspects PACS96:05.45.+b Theory and models of chaotic systems 

License:  Creative Commons: AttributionNoncommercialNo Derivative Works 3.0  
Additional Information:  PACS96:03.65.Sq Semiclassical theories and applications PACS96:03.20.+i Classical mechanics of discrete systems: general mathematical aspects PACS96:05.45.+b Theory and models of chaotic systems 

URI:  http://tuprints.ulb.tudarmstadt.de/id/eprint/1130  
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