Spectral Approach to Chaos and Quantum-Classical Correspondence in   Quantum Mapas
  Correspondence in quantum chaotic systems is lost in short time scales. Introducing some noise we study the spectrum of the resulting coarse grained propagaor of density matrices. Some differen methods to compute the spectrum are reviewed. Moreover, the relationship between the eigenvalues of the coarse-grained superoperator and the classical Ruelle-Pollicott resonances is remarked. As a concequence, classical decay rates in quantum time dependent quantities appear. 
Ignacio Garcia-Mata and Marcos Saraceno