Semiclassical Formulae For Wigner Distributions

TL;DR

This paper explores the mathematical and physical implications of Ruelle resonances in chaotic systems, connecting them with semiclassical zeta functions and quantum mechanics, and providing numerical results for scattering systems.

Contribution

It introduces a new correspondence between weighted and semiclassical zeta functions and offers a high frequency interpretation of Ruelle distributions in negative curvature.

Findings

Invariant Ruelle distributions as residues of zeta functions

High frequency interpretation as quantum matrix coefficients

Numerical phase space distributions in 3-disk scattering

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.