Aspects of Semiclassical Theory in the Presence of Classical Chaos

Topics concerning the rigorous validity of semiclassical approximations in the presence of classical chaos are discussed from three viewpoints: time-evolution (how long semiclassical behavior can be ascertained in the propagator, with the use of reduction methods and quantum maps), eigenvalues (emphasizing the importance of the two separate divergences in trace formulas, the classical divergence from chaos and the quantum divergence from ℏ-corrections), and eigenfunctions (which display their semiclassical behavior most explicitly in a “stellar” representation over the phase space). All areas are still characterized by a scarcity of proven results and of mathematical investigation tools against an increasing set of observations.