Toward a comprehensive semiclassical ergodic theory

We identify quantum dynamical conditions that tend, as ℏ→0; to the classical conditions of ergodicity, weak mixing, mixing, and absolutely continuous spectrum. We use these conditions to derive asymptotic properties of wave functions, matrix elements, and energy levels of systems with various ergodic properties in the classical limit. Some of these properties are new while others modify older predictions of ‘‘quantum ergodic properties.’’ Thus, the present work serves to extend, unify, and refine a number of previous theories. One new prediction is that weak mixing leads to the Van Hove diagonal singularity condition for a certain class of operators. This result may have significant implications for the applicability of nonequilibrium statistical mechanics to finite systems.