Heidelberg 1999 – wissenschaftliches Programm

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MP: Theoretische und Mathematische Grundlagen der Physik

MP 3: Hauptvortr
äge III

MP 3.3: Hauptvortrag

Donnerstag, 18. März 1999, 11:55–12:45, MA1

Quantum mechanics of chaotic maps — •Jens Marklof — Institut des Hautes Etudes Scientifique, Bures-sur-Yvette, France and Laboratoire de Physique Théorique et Modèles Statistiques, Orsay, France

In the theory of quantum chaos one is concerned with the quantization of chaotic Hamiltonian systems, and their stochastic properties in the semiclassical limit. Some of the simplest classical Hamiltonian systems are the symplectic maps of the 2-torus, which can exhibit all kinds of dynamics — from complete integrability to strong chaos.

I will explain how these maps can be quantized, and how the chaoticity of the classical system emerges in the semiclassical limit. Note that the latter point is a highly non-trivial one, since the semiclassical limit does not commute with the time-to-infinity limit used to characterize chaos in classical mechanics.

The main focus will be on quantum ergodicity, which describes the asymptotic distribution of semiclassical eigenstates. More precisely, this means that expectation values of quantum observables converge for almost all eigenstates to the phase-space average of the classical observable, if the classical system is ergodic (Shnirelman’s Theorem). I will also present recent examples of quantum unique ergodicity (QUE), i.e., systems for which the above property holds for all semiclassical eigenstates. These systems are the first known examples of QUE (joint work with Zeev Rudnick, Tel Aviv University).