Quantum toral automorphisms (Pär Kurlberg, KTH, Stockholm)

*    Classical and quantum mechanics on the torus

    Introduction to classical and quantum dynamics on the torus.
    What does "quantization" mean, and what important properties
    should it have?  The quantum version of the classical state of
    a particle (i.e., knowing its position and momentum) is a
    vector in a certain Hilbert space.  Classical observables
    (such as measuring position and momentum) are quantized by
    associating certain linear operators; the result of a
    measurement is then an eigenvalue of said operator, and the
    new state of the system after the measurement corresponds to
    an eigenvector of the operator.  The time evolution of a
    quantum system, called the quantum propagator, is given by a
    unitary operator acting on the Hilbert space of states.  Since
    classical mechanics should be a limiting case of quantum
    mechanics (as Planck's constant tends to zero), certain
    compatibility requirements between the quantized observables
    and the quantum propagator must be satisfied.  We will show
    how to do this for toral automorphisms, also known as "CAT

*    Quantum ergodicity

    Quantum ergodicity is a counter-part to classical ergodicity,
    namely that *most* eigenfunctions are "uniformly spread out"
    in a certain sense.  This is often known as Schnirelman's
    theorem; we will give Zelditch's proof of this by computing
    the variance of diagonal matrix coefficients, with respect to
    a basis of eigenfunctions of the quantum propagator.  However,
    "most" does not mean all - maybe there potential
    counterexamples to equidistribution!?

*    Arithmetic quantum unique ergodicicy

    In general, quantized cat maps can have large spectral
    degeneracies, so there is a lot of lee-way to form "nasty"
    linear combination of eigenfunctions in a fixed eigenspace.
    However, massive amounts of degeneracies is often coupled with
    the existence of large families of symmetries.  By looking at
    maximally desymmetrized eigenfunctions ("Hecke
    eigenfunctions"), it turns out that the degeneracies can be
    controlled, and that it is possible to show that *all* Hecke
    eigenfunctions are equidistributed.

*    Properties of Hecke eigenfunctions

    We will study properties, such as value distribution and
    L^p-norms, of Hecke eigenfunctions.  The main tool here is the
    theory of exponential sums over finite fields, in particular
    the "Riemann hypothesis for function fields."

*    Quantum scarring for special eigenfunctions

    In case of *massive* spectral degeneracies, Faure,
    Nonnenmacher, and de Bievre has shown that it is possible to
    produce a subsequence of eigenfunctions that are *not*
    equidistributed - so called "quantum scars".  The main idea
    here is to pull back "squeezed coherent states" to the plane,
    analyze their time evolution in the plane, then project them
    back to the torus; this coupled with a detailed analysis of
    the Diophantine properties of the stable/unstable manifolds
    can then be used to produce eigenfunctions that scar by taking
    (very short) time averages of these squeezed coherent states.