Fractal Weyl laws in discrete models of chaotic scattering

Abstract

We analyze simple models of quantum chaotic scattering, namely quantized open baker's maps. We numerically compute the density of quantum resonances in the semiclassical régime. This density satisfies a fractal Weyl law, where the exponent is governed by the (fractal) dimension of the set of trapped trajectories. This type of behaviour is also expected in the (physically more relevant) case of Hamiltonian chaotic scattering. Within a simplified model, we are able to rigorously prove this Weyl law, and compute quantities related to the "coherent transport" through the system, namely the conductance and "shot noise". The latter is close to the prediction of random matrix theory.

Invited article in the Special Issue of Journal of Physics A on "Trends in Quantum Chaotic Scattering"

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Cited by in corpus (41)

• Leaking Chaotic Systems
• Semiclassical structure of chaotic resonance eigenfunctions
• Spectral problems in open quantum chaos
• Quantum-to-classical correspondence in open chaotic systems
• Resonances in open quantum maps
• Nodal portraits of quantum billiards: Domains, lines, and statistics
• Weyl asymptotics: From closed to open systems
• Distribution of resonances for open quantum maps
• Localization of resonance eigenfunctions on quantum repellers
• Quantum chaotic resonances from short periodic orbits
• Fractal Weyl laws for quantum decay in generic dynamical systems
• Scarring in open quantum systems
• Resonances for open quantum maps and a fractal uncertainty principle
• Lifetime statistics in chaotic dielectric microresonators
• Hierarchical Fractal Weyl Laws for Chaotic Resonance States in Open Mixed Systems
• Resonance distribution in open quantum chaotic systems
• From open quantum systems to open quantum maps
• On the resonance eigenstates of an open quantum baker map
• Resonance eigenfunction hypothesis for chaotic systems
• Short periodic orbit approach to resonances and the fractal Weyl law
• Distribution of resonances in the quantum open baker map
• On the mean density of complex eigenvalues for an ensemble of random matrices with prescribed singular values
• Formation and interaction of resonance chains in the open 3-disk system
• Faster than expected escape for a class of fully chaotic maps
• Nodal domains in open microwave systems
• Quantum chaos in the spectrum of operators used in Shor's algorithm
• Multifractality of open quantum systems
• Classical transients and the support of open quantum maps
• Universal single-mode lasing in fully chaotic two-dimensional microcavity lasers under continuous-wave operation with large pumping power
• Using the Hadamard and related transforms for simplifying the spectrum of the quantum baker's map
• Transient features of quantum open maps
• Weyl law for open systems with sharply divided mixed phase space
• Semiclassical theory of chaotic quantum resonances
• Resonance spectra for quantum maps of kicked scattering systems by complex scaling
• The modular multiplication operator and the quantized bakers maps
• Resonance states of the three-disk scattering system
• Spectral behavior of contractive noise
• Counting statistics of chaotic resonances at optical frequencies: theory and experiments
• Weyl law for fat fractals
• Resonances in the Two-Centers Coulomb Systems
• Distribution of resonances for hyperbolic surfaces