CHAOTIC EIGENFUNCTIONS IN PHASE-SPACE

We study individual eigenstates of quantized area-preserving maps on t he 2-torus which are classically chaotic. In order to analyze their se miclassical behavior, we use the Bargmann-Husimi representations for q uantum states as well as their stellar parametrization, which encodes states through a minimal set of points in phase space (the constellati on of zeros of the Husimi density). We rigorously prove that a semicla ssical uniform distribution of Husimi densities on the torus entails a similar equidistribution for the corresponding constellations. We ded uce from this property a universal behavior for the phase patterns of chaotic Bargmann eigenfunctions which is reminiscent of the WKB approx imation for eigenstates of integrable systems (though in a weaker sens e). In order to obtain more precise information on ''chaotic eigencons tellations,'' we then model their properties by ensembles of random st ates, generalizing former results on the 2-sphere to the torus geometr y. This approach yields statistical predictions for the constellations which fit quite well the chaotic data. We finally observe that specif ic dynamical information, e.g., the presence of high peaks (like scars ) in Husimi densities, can be recovered from the knowledge of a few lo ng-wavelength Fourier coefficients, which therefore appear as valuable order parameters at the level of individual chaotic eigenfunctions.