Quantum Ergodicity aims at understanding the eigenstates of quantum mechanical systems admitting chaotic classical limiting dynamics. A paradigmatic system is the Laplace-Beltrami operator on a compact manifold of negative sectional curvature: its classical limit is the geodesic flow on the manifold, which is of Anosov type. Although no explicit expression is available for the eigenstates, one may use various tools from semiclassical analysis in order to gather some partial information on their structure. The central result (Quantum Ergodicity Theorem) states that almost all eigenstates are equidistributed over the energy shell, in the semiclassical limit, provided the classical system is ergodic. The lectures review the background techniques of semiclassical analysis and ergodic theory, give several versions of the QE theorem, and present several extensions of the result, which apply to specific systems, for instance chaotic systems enjoying arithmetic symmetries.
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Ulrich Bunke, Stéphane Nonnenmacher, Roman Schubert, Arbeitsgemeinschaft: Quantum Ergodicity. Oberwolfach Rep. 8 (2011), no. 4, pp. 2781–2835DOI 10.4171/OWR/2011/49