Semiclassical Formulae For Wigner Distributions

Abstract

In this paper we give an overview over some aspects of the modern mathematical theory of Ruelle resonances for chaotic, i.e. uniformly hyperbolic, dynamical systems and their implications in physics. First we recall recent developments in the mathematical theory of resonances, in particular how invariant Ruelle distributions arise as residues of weighted zeta functions. Then we derive a correspondence between weighted and semiclassical zeta functions in the setting of negatively curved surfaces. Combining this with results of Hilgert, Guillarmou and Weich yields a high frequency interpretation of invariant Ruelle distributions as quantum mechanical matrix coefficients in constant negative curvature. We finish by presenting numerical calculations of phase space distributions in the more physical setting of 3-disk scattering systems.

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References in corpus (21)

• Ruelle-Perron-Frobenius spectrum for Anosov maps
• Fractal Weyl laws for chaotic open systems
• Quantum-to-classical crossover of quasi-bound states in open quantum systems
• Symmetry Decomposition of Chaotic Dynamics
• Upper bound on the density of Ruelle resonances for Anosov flows
• Quantum Mechanics and Semiclassics of Hyperbolic n-Disk Scattering Systems
• A smooth introduction to the wavefront set
• Experimental Observation of the Spectral Gap in Microwave n-Disk Systems
• Weyl asymptotics: From closed to open systems
• Frobenius-Perron Resonances for Maps with a Mixed Phase Space
• Power spectrum of the geodesic flow on hyperbolic manifolds
• Localization of resonance eigenfunctions on quantum repellers
• Resonances of the Frobenius-Perron Operator for a Hamiltonian Map with a Mixed Phase Space
• Patterson-Sullivan distributions and quantum ergodicity
• Classical versus Quantum Time Evolution of Densities at Limited Phase-Space Resolution
• Quantum resonances and decay of a chaotic fractal repeller observed using microwaves
• Formation and interaction of resonance chains in the open 3-disk system
• The Ruelle zeta function at zero for nearly hyperbolic 3-manifolds
• Wavefunctions, Green's functions and expectation values in terms of spectral determinants
• Quantization of Classical Maps with tunable Ruelle-Pollicott Resonances
• Numerical resonances for Schottky surfaces via Lagrange-Chebyshev approximation