Fractal Weyl law for quantum fractal eigenstates

TL;DR

This paper demonstrates that the number and distribution of quantum resonant states in a chaotic system follow a fractal Weyl law, linking quantum properties to classical fractal structures.

Contribution

It provides numerical evidence that quantum resonant states adhere to a fractal Weyl law and connects their Husimi distributions to classical strange repellers.

Findings

Number of resonant states follows fractal Weyl law

Husimi distributions align with classical strange repellers

Escape rate distribution converges to a fixed profile

Abstract

The properties of the resonant Gamow states are studied numerically in the semiclassical limit for the quantum Chirikov standard map with absorption. It is shown that the number of such states is described by the fractal Weyl law and their Husimi distributions closely follow the strange repeller set formed by classical orbits nonescaping in future times. For large matrices the distribution of escape rates converges to a fixed shape profile characterized by a spectral gap related to the classical escape rate.