Weyl law for open systems with sharply divided mixed phase space

TL;DR

This paper extends the Weyl law to systems with sharply divided mixed phase space, incorporating regular islands and sticky regions, supported by numerical validation on piecewise linear maps.

Contribution

It introduces a generalized Weyl law for mixed phase space systems, accounting for regular islands and sticky regions, with numerical evidence supporting its validity.

Findings

Good agreement with numerical data for sharply divided phase space

Validation also for systems with tiny island chains

Discovery of a power law exponent in classical escape probabilities

Abstract

A generalization of the Weyl law to systems with a sharply divided mixed phase space is proposed. The ansatz is composed of the usual Weyl term which counts the number of states in regular islands and a term associated with sticky regions in phase space. For a piecewise linear map, we numerically check the validity of our hypothesis, and find good agreement not only for the case with a sharply divided phase space, but also for the case where tiny island chains surround the main regular island. For the latter case, a non-trivial power law exponent appears in the survival probability of classical escaping orbits, which may provide a clue to develop the Weyl law for more generic mixed systems.