Quantum Mechanics and Semiclassics of Hyperbolic n-Disk Scattering Systems

Abstract

The scattering problems of a scalar point particle from a finite assembly of n>1 non-overlapping and disconnected hard disks, fixed in the two-dimensional plane, belong to the simplest realizations of classically hyperbolic scattering systems. 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. 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 m-th quantum cumulant is the m-th 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.

131 pages (LaTeX), 47 figures (PostScript), uses elsart.sty, epsfig.sty, epsf.sty and epsf.tex; Habilitationsschrift (TU Darmstadt), see http://crunch.ikp.physik.tu-darmstadt.de/~wirzba/Habilitation.html

References in corpus (11)

• Periodic Orbit Theory of Diffraction
• Symmetry Decomposition of Chaotic Dynamics
• Density-of-States Calculations and Multiple-Scattering Theory for Photons
• The Role of Diffraction in the Quantization of Dispersing Billiards
• Entire Fredholm determinants for Evaluation of Semi-classical and Thermodynamical Spectra
• Resummation of classical and semiclassical periodic orbit formulas
• Application of the diffraction trace formula to the three disk scattering system
• Inside-Outside Duality for Planar Billiards -- A Numerical Study
• Quantum Fluids and Classical Determinants
• Three disks in a row: A two-dimensional scattering analog of the double-well problem
• Inclusion of Diffraction Effects in the Gutzwiller Trace Formula

Cited by in corpus (47)

• Scattering Theory Approach to Electrodynamic Casimir Forces
• Casimir forces in a T operator approach
• Scalar Casimir effect between Dirichlet spheres or a plate and a sphere
• Fractal Weyl laws for chaotic open systems
• Non-equilibrium Fluctuational Quantum Electrodynamics: Heat Radiation, Heat Transfer, and Force
• Use of Harmonic Inversion Techniques in Semiclassical Quantization and Analysis of Quantum Spectra
• Fractal Weyl laws in discrete models of chaotic scattering
• Spectral problems in open quantum chaos
• Numerical Study of Quantum Resonances in Chaotic Scattering
• Experimental Observation of the Spectral Gap in Microwave n-Disk Systems
• Resonances in open quantum maps
• Distribution of resonances for open quantum maps
• Casimir Interaction among Objects Immersed in a Fermionic Environment
• Decimation and Harmonic Inversion of Periodic Orbit Signals
• Billiard Systems in Three Dimensions: The Boundary Integral Equation and the Trace Formula
• Quantum chaotic resonances from short periodic orbits
• The Casimir effect as scattering problem
• Casimir interaction between normal or superfluid grains in the Fermi sea
• Fractal Weyl law for three-dimensional chaotic hard-sphere scattering systems
• Statistics of Chaotic Resonances in an Optical Microcavity
• Surface polaritons on left-handed cylinders: A complex angular momentum analysis
• From open quantum systems to open quantum maps
• Formation and interaction of resonance chains in the open 3-disk system
• Semiclassical Quantization by Pade Approximant to Periodic Orbit Sums
• Universal intensity statistics of multifractal resonance states
• Chaotic resonance modes in dielectric cavities: Product of conditionally invariant measure and universal fluctuations
• Geometry and material effects in Casimir physics - Scattering theory
• Periodic orbit theory for the Hénon-Heiles system in the continuum region
• Semiclassical Formulae For Wigner Distributions
• Something Can Come of Nothing: Surface Approaches to Quantum Fluctuations and the Casimir Force
• Local random vector model for semiclassical fractal structure of chaotic resonance states
• Multiple scattering model of the quantum random Lorentz gas
• Resonance chains in open systems, generalized zeta functions and clustering of the length spectrum
• Traces and determinants of strongly stochastic operators
• Resonance states of the three-disk scattering system
• Wave Chaos in Elastodynamic Cavity Scattering
• Quantum Vacuum Energy of Self-Similar Configurations
• Superiority of semiclassical over quantum mechanical calculations for a three-dimensional system
• Short-Pulsed Wavepacket Propagation in Ray-Chaotic Enclosures
• Periodic orbit quantization of chaotic systems with strong pruning
• Exact Electromagnetic Casimir Energy of a Disk Opposite a Plane
• Classical, semiclassical, and quantum investigations of the 4-sphere scattering system
• Towards a semiclassical understanding of chaotic single- and many-particle quantum dynamics at post-Heisenberg time scales
• Families of spherical caps: spectra and ray limit
• Wave packet autocorrelation functions for quantum hard-disk and hard-sphere billiards in the high-energy, diffraction regime
• Analytic continuation of the resolvent of the Laplacian and the dynamical zeta function
• Möbius transformations and the configuration space of a Hilbert snake