Short periodic orbit approach to resonances and the fractal Weyl law

TL;DR

This paper demonstrates that the short periodic orbit approach effectively reduces the eigenvalue problem's dimensionality for open quantum maps, aligning with the fractal Weyl law and accurately reproducing phase space projectors.

Contribution

It provides numerical evidence that the short periodic orbit method applies to open quantum maps, confirming the fractal Weyl law and reproducing phase space projectors.

Findings

Eigenvalue problem dimensionality reduces according to the fractal Weyl law.

Method accurately reproduces phase space projectors on the classical repeller.

Numerical validation using open baker and cat maps.

Abstract

We investigate the properties of the semiclassical short periodic orbit approach for the study of open quantum maps that was recently introduced in [M. Novaes, J.M. Pedrosa, D. Wisniacki, G.G. Carlo, and J.P. Keating, Phys. Rev. E 80, 035202(R) 2009]. We provide conclusive numerical evidence, for the paradigmatic systems of the open baker and cat maps, that by using this approach the dimensionality of the eigenvalue problem is reduced according to the fractal Weyl law. The method also reproduces the projectors $|\psi^R_n><\psi_n^L|$, which involves the right and left states associated with a given eigenvalue and is supported on the classical phase space repeller.