Summer school on Quantum Chaos: some
Abstracts
- Zeév Rudnick - Introduction to Quantum Chaos.
- Stéphane Nonnenmacher + Gabriel Rivière - Introduction to Dynamical systems (4h)
The quantitative statistical properties are characterized by invariant (probability) measures. The study of these measures, and of the associated long time properties (ergodicity, mixing), is the object of ergodic theory.
We will mainly focus on "chaotic" systems, of which we will present some model examples. "Chaos" is usually associated with the local unstability of the orbits (hyperbolicity), but this local property often implies global ergodic properties, as well as constraints on the complexity of the dynamics.
In the second part of the lectures, we will introduce the notion of Kolmogorov-Sinai entropy which gives information on the intrinsic complexity of a measurable dynamical system. We will compute it for several examples of dynamical systems relevant to quantum chaos. We will also describe basic properties of this quantity. Finally, we will discuss the notions of topological pressure, and its connection with Kolmogorov-Sinai entropy.
A classical (and very complete) reference for these notions is P.Walter's Introduction to ergodic theory, as well as the Introduction to the modern theory of dynamical systems by B.Hasselblatt and A.Katok.
- Andrew Hassell - Semiclassical Analysis and Quantum Ergodicity (4h)
The slides of the lectures are available here.
- Jonathan Keating - Random Matrix Theory and Quantum Chaos (2h)
- Mikhail Sodin - Nodal portraits of random waves (2h)
I plan to explain how basic results from the ergodic theory help to
find the order of growth of the number of components of zero sets of
smooth Gaussian random functions of several real variables.
The slides of the lectures are available here.
- Sebastian Müller - Spectral vs. periodic orbit correlations
I will discuss recent research to understand why chaotic quantum
systems display universal spectral statistics, in agreement with predictions from random matrix theory. In the semiclassical limit correlations between the quantum mechanical energy
levels of a chaotic system are closely connected to correlations between its classical periodic orbits. The key ingredient to identify such correlations are self-encounters where different parts of a periodic orbit come close to each other. By changing the connections inside these self-encounters one obtains bunches of orbits that are very similar to each other. In the semiclassical approach contributions from different periodic orbits in a bunch can interfere
constructively. I will discuss how this interference gives rise to universal spectral statistics.
I will also sketch how the semiclassical approach has to be modified to access more elusive aspects of spectral statistics (the behaviour of the spectral for factor for large times).
An introduction into this line of research is given in
S. Müller and M. Sieber, Quantum Chaos and Quantum Graphs,
The Oxford Handbook of Random Matrix Theory, eds. G. Akemann, J. Baik, and P. Di
Francesco (2011)
The slides of the lecture are available here.
- Jens Marklof - Spectral correlations for integrable
systems
- Manfred Einsiedler - Arithmetic Quantum Unique Ergodicity
Notes relevant for the lecture can be found here.
- Steve Zelditch - Ergodicity and nodal sets of eigenfunctions
It is conjectured that ergodicity of the geodesic flow implies
that nodal sets of eigenfunctions become uniformly distributed as the eigenvalue
tends to infinity. This conjecture is far out of reach, but it is possible to prove the
analogous statement in the complex domain, i.e. if one analytically continues eigenfunctions
to the complexification of a real analytic manifold and considers the complex zeros. It is
also possible to obtain the distributions of intersections of (complex) geodesics and
nodal sets.
- John Toth - Eigenfunction restriction bounds
quantum integrable and ergodic cases as well as for eigenfunction perturbations.
The slides of the lecture are available here.
- Andrew Hassell - Non-QUE for partially rectangular domains
- Gabriel Rivière - Entropic bounds for eigenfunctions
We will discuss results on the localization properties of eigenfunctions of the Laplacian on manifolds of negative curvature. More precisely, we will explain how one can obtain lower bounds for the Kolmogorov Sinai entropy of the corresponding semiclassical measures.
The (handwritten) notes of the lecture can be found here.
- Zeév Rudnick - Pseudo-integrable systems
- Stéphane Nonnenmacher - Quantum chaos with open systems
I plan to present "open" quantized chaotic systems and the associated spectral problems. Such systems include scattering systems, such that the classical set of trapped orbits ("trapped set") supports a chaotic flow. In that case, the spectrum of the Hamiltonian is continuous, but also features complex valued resonances, associated with metastable states. One objective of "open quantum chaos" is to characterize this (nonselfadjoint) resonance spectrum, as well as the phase space structure of the associated states.
In this context, I will mention Fractal Weyl Laws, which connect the resonance counting with the fractal dimension of the trapped set. I will also explain the connection between the classical hyperbolicity and a resonance gap.
The slides of the lecture are available here.