Dresden 2000 – wissenschaftliches Programm

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SYMA: Mathematical Aspects of Dynamical Systems

SYMA IV: HV IV

SYMA IV.1: Hauptvortrag

Montag, 20. März 2000, 12:15–12:55, H 04

Microlocal Analysis and the Semiclassical Limit in Quantum Mechanics — •Roman Schubert — Abteilung Theoretische Physik, Universität Ulm, Albert-Einstein-Allee 11, 89069 Ulm

Microlocal analysis, which was originally developed for studying partial differential operators, has now found many applications in quantum mechanics. We will give an introduction into microlocal analysis from the point of view of semiclassical quantum mechanics, and then turn to applications in quantum chaos.

The main focus will be on how the eigenfunctions of a Hamilton operator reflect the properties of the flow generated by the corresponding classical Hamilton function. In case that the flow is ergodic, the quantum ergodicity theorem states that almost all eigenfunctions become equidistributed in a suitable sense. On the other hand, the presence of invariant tori, or elliptic periodic orbits in the classical system allows the construction of approximate solutions of the stationary Schrödinger equation, so called quasimodes. A third method by which localisation properties of eigenfunctions can be studied is the use of approximate projection operators onto invariant sets in phase space.

Finally, the general picture of the semiclassical properties of eigenfunctions which emerges from these results and some open problems will be discussed.