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

maps".

* 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.