Anatomy of Quantum Chaotic Eigenstates

We present one aspect of “Quantum Chaos”, namely the description of high frequency eigenmodes of a quantum system, the classical limit of which is chaotic (we call such eigenmodes “chaotic eigenmodes”). A paradigmatic example is provided by the eigenmodes of the Laplace–Beltrami operator on a compact Riemannian manifold of negative curvature: the corresponding classical dynamics is the geodesic flow on the manifold, which is strongly chaotic (Anosov). Other well-studied classes of examples include certain Euclidean domains (“chaotic billiards”), or quantized chaotic symplectomorphisms of the two-dimensional torus.

We propose several levels of description, some of them allowing for mathematical rigor, others being more heuristic.

The macroscopic distribution of the eigenstates makes use of semiclassical measures, which are probability measures invariant w.r.t. the classical dynamics; these measures reflect the asymptotic “shape” of a sequence of high frequency eigenmodes. The quantum ergodicity theorem states that the vast majority of the eigenstates is associated with the “flat” (Liouville) measure. A major open problem addresses the existence of “exceptional” eigenmodes admitting different macroscopic properties.

The microscopic description deals with the structure of the eigenfunctions at the scale of their wavelengths. It is mainly of statistical nature: it addresses, for instance, the value distribution of the eigenfunctions, their shortdistance correlation functions, the statistics of their nodal sets or domains. This microscopic description mostly relies on a random state (or random wave) Ansatz for the chaotic eigenmodes, which is far from being mathematically justified, but already offers interesting challenges for probabilists and harmonic analysts.