Quantum Chaos and Quantum Ergodicity

We report on some of our results which have been achieved within the DFG Schwerpunktprogramm “Ergodentheorie, Analysis und effiziente Simulation dynamischer Systeme” (1994-2000).One main point of our research programme has been the search for universal statistical properties of energy spectra and eigen-functions of quantum mechanical systems whose classical dynamics is chaotic. The mode-fluctuation distribution P(W)has been proposed as a universal signature of quantum chaos and a conjecture on its limit distribution has been put forward. The conjecture turns out to be mathematically equivalent to a hypothesis on the value distribution of dynamical zeta functions on the critical line and has been successfully tested for several chaotic systems. For certain systems this can be expressed in terms of the Selberg zeta function. For a large class of ergodic systems the quantum ergodicity theorem holds, which (roughly speaking) states that almost all eigenfunctions become equidistributed in the semiclassical limit. Particular attention has been paid to the question of subsequences of exceptional, non-quantum ergodic eigenfunctions, and their counting function. Such eigenfunctions are for example bouncing-ball modes occurring in billiards with two parallel walls (like the stadium or the Sinai billiard). Also “scarred” eigenfunctions showing localization along unstable periodic orbits could give rise to a non-quantum ergodic subsequence of eigenfunctions. Furthermore, the rate by which the classical limit is approached has been studied. We conclude by giving a short summary of the other topics studied by our group within the Schwerpunktprogramm.