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Unformatted text preview: S´ eminaire Poincar´ e XIV (2010) 177 – 220 S´ eminaire Poincar´ e Anatomy of quantum chaotic eigenstates St´ ephane Nonnenmacher * Institut de Physique Th´ eorique CEA-Saclay 91191 Gif-sur-Yvette, France 1 Introduction These notes present a description of quantum chaotic eigenstates , that is bound states of quantum dynamical systems, whose classical limit is chaotic. The classical dynamical systems we will be dealing with are mostly of two types: geodesic flows on Euclidean domains (“billiards”) or compact riemannian manifolds, and canonical transformations on a compact phase space; the common feature is the “chaoticity” of the dynamics. The corresponding quantum systems will always be considered within the semiclassical (or high-frequency) r´ egime, in order to establish a connection them with the classical dynamics. As a first illustration, we plot below two eigenstates of a paradigmatic system, the Laplacian on the stadium billiard , with Dirichlet boundary conditions 1 . The study of chaotic eigenstates makes up a large part of the field of quan- tum chaos . It is somewhat complementary with the contribution of J. Keating (who will focus on the statistical properties of quantum spectra, another major topic in quantum chaos). I do not include the study of eigenstates of quantum graphs (a recent interesting development in the field), since this question should be addressed in U.Smilansky’s lecture. Although these notes are purely theoretical, H.-J. St¨ock- mann’s lecture will show that the questions raised have direct experimental appli- cations (his lecture should present experimentally measured eigenmodes of 2- and 3-dimensional “billiards”). One common feature of the chaotic eigenfunctions (except in some very specific systems) is the absence of explicit, or even approximate, formulas. One then has to resort to indirect, rather unprecise approaches to describe these eigenstates. We will use various analytic tools or points of view: deterministic/statistical, macro/microscopic, pointwise/global properties, generic/specific systems. The level of rigour in the results varies from mathematical proofs to heuristics, generally sup- ported by numerical experiments. The necessary selection of results reflects my per- sonal view or knowledge of the subject, it omits several important developments, and is more “historical” than sharply up-to-date. The list of references is thick, but in no way exhaustive. * I am grateful to E.Bogomolny, who allowed me to reproduce several plots from . The author has been partially supported by the Agence Nationale de la Recherche under the grant ANR-09-JCJC-0099-01. These notes were written while he was visiting the Institute of Advanced Study in Princeton, supported by the National Science Foundation under agreement No. DMS-0635607....
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