Short periodic orbit approach to resonances and the fractal Weyl law

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.