Fractal Weyl law for three-dimensional chaotic hard-sphere scattering systems

TL;DR

This paper tests the fractal Weyl law in a three-dimensional chaotic scattering system, specifically the four-sphere billiard, by analyzing the chaotic repeller and applying semiclassical quantization methods.

Contribution

It provides the first empirical test of the fractal Weyl law for a 3D open scattering system using cycle expansion and symmetry analysis.

Findings

Confirmed the fractal Weyl law for various symmetry subspaces.

Analyzed the chaotic repeller in the four-sphere billiard.

Discussed semiclassical quantization via cycle expansion.

Abstract

The fractal Weyl law connects the asymptotic level number with the fractal dimension of the chaotic repeller. We provide the first test for the fractal Weyl law for a three-dimensional open scattering system. For the four-sphere billiard, we investigate the chaotic repeller and discuss the semiclassical quantization of the system by the method of cycle expansion with symmetry decomposition. We test the fractal Weyl law for various symmetry subspaces and sphere-to-sphere separations.