Fractal Weyl laws in discrete models of chaotic scattering

TL;DR

This paper studies quantum chaotic scattering using open baker's maps, demonstrating a fractal Weyl law for resonance density, supported by numerical and rigorous results, and analyzing conductance and shot noise consistent with random matrix theory.

Contribution

It provides the first rigorous proof of the fractal Weyl law in a simplified quantum chaotic scattering model and connects resonance density to fractal trapped sets.

Findings

Resonance density follows a fractal Weyl law governed by trapped set dimension.

Numerical computations confirm the law in quantum open baker's maps.

Conductance and shot noise match predictions from random matrix theory.

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 r\'{e}gime. 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.