Spectral problems in open quantum chaos

Abstract

This review article will present some recent results and methods in the study of 1-particle quantum or wave scattering systems, in the semiclassical/high frequency limit, in cases where the corresponding classical/ray dynamics is chaotic. We will focus on the distribution of quantum resonances, and the structure of the corresponding metastable states. Our study includes the toy model of open quantum maps, as well as the recent quantum monodromy operator method.

Compared with the previous version, misprints and typos have been corrected, and the bibliography updated

References in corpus (11)

• Fractal Weyl laws for chaotic open systems
• Quantum-to-classical crossover of quasi-bound states in open quantum systems
• Semiclassical structure of chaotic resonance eigenfunctions
• Microlocal analysis of asymptotically hyperbolic and Kerr-de Sitter spaces
• Fractal Weyl law for chaotic microcavities: Fresnel's laws imply multifractal scattering
• Fractal Weyl law for quantum fractal eigenstates
• Trace formula for dielectric cavities II: Regular, pseudo-integrable, and chaotic examples
• Wave decay on convex co-compact hyperbolic manifolds
• Fractal Weyl law for three-dimensional chaotic hard-sphere scattering systems
• On the resonance eigenstates of an open quantum baker map
• On the resonances of convex co-compact subgroups of arithmetic groups

Cited by in corpus (33)

• Leaking Chaotic Systems
• Spectral gaps without the pressure condition
• Random Lindblad Dynamics
• Resonance projectors and asymptotics for r-normally hyperbolic trapped sets
• Experimental Observation of the Spectral Gap in Microwave n-Disk Systems
• Resonances in open quantum maps
• Fourier dimension and spectral gaps for hyperbolic surfaces
• An introduction to fractal uncertainty principle
• Resonances for open quantum maps and a fractal uncertainty principle
• Formation and interaction of resonance chains in the open 3-disk system
• Dolgopyat's method and the fractal uncertainty principle
• Short periodic orbits theory for partially open quantum maps
• Classical transients and the support of open quantum maps
• Transient features of quantum open maps
• Improved fractal Weyl bounds for hyperbolic manifolds
• Weyl law for open systems with sharply divided mixed phase space
• The role of short periodic orbits in quantum maps with continuous openings
• Resonance chains and geometric limits on Schottky surfaces
• Resonance spectra for quantum maps of kicked scattering systems by complex scaling
• Resonance chains in open systems, generalized zeta functions and clustering of the length spectrum
• Asymptotic spectral gap and Weyl law for Ruelle resonances of open partially expanding maps
• Resonance states of the three-disk scattering system
• Resolvent estimates on asymptotically cylindrical manifolds and on the half line
• Fractal Weyl laws for asymptotically hyperbolic manifolds
• Spectral gap for obstacle scattering in dimension 2
• Semiclassical Limit of Resonance States in Chaotic Scattering
• Spectral behavior of contractive noise
• Mathematical Study of Scattering Resonances
• Resolvent estimates in strips for obstacle scattering in 2D and local energy decay for the wave equation
• Fractal Weyl laws and wave decay for general trapping
• Localization in chaotic systems with a single-channel opening
• Weyl law for contractive maps
• Scattering resonances of large weakly open quantum graphs