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 [Novaes, Pedrosa, Wisniacki, Carlo, and Keating, Phys. Rev. E 80, 035202(R) (2009)]. We provide solid 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 |ψ(n)(R)><ψ(n)(L)|, which involves the right and left states associated with a given eigenvalue and is supported on the classical phase-space repeller.

Quantum chaotic resonances from short periodic orbits.

We present an approach to calculating the quantum resonances and resonance wave functions of chaotic scattering systems, based on the construction of states localized on classical periodic orbits and adapted to the dynamics. Typically only a few such states are necessary for constructing a resonance. Using only short orbits (with periods up to the Ehrenfest time), we obtain approximations to the longest-living states, avoiding computation of the background of short living states. This makes our approach considerably more efficient than previous ones. The number of long-lived states produced within our formulation is in agreement with the fractal Weyl law conjectured recently in this setting. We confirm the accuracy of the approximations using the open quantum baker map as an example.