Phase space localization of chaotic eigenstates: Violating ergodicity
  The correlation between level velocities and eigenfunction intensities provides a new way of exploring phase space localization in quantized non-integrable systems. It can also serve as a measure of deviations from ergodicity due to quantum effects for typical observables. This paper relies on two well known paradigms of quantum chaos, the bakers map and the standard map, to study correlations in simple, yet chaotic, dynamical systems. The behaviors are dominated by the presence of several classical structures. These primarily include short periodic orbits and their homoclinic excursions. The dependences of the correlations deriving from perturbations allow for eigenfunction features violating ergodicity to be selectively highlighted. A semiclassical theory based on periodic orbit sums leads to certain classical correlations that are super-exponentially cut off beyond a logarithmic time scale. The theory is seen to be quite successful in reproducing many of the quantum localization features. 
Arul Lakshminarayan, Nicholas R. Cerruti and Steven Tomsovic