KURLBERG:
"On the distribution of matrix elements for the quantum cat map"
For many classically chaotic systems it is believed that the quantum
wave functions become uniformly distributed, that is the matrix
elements of smooth observables tend to the phase space average of the
observable. We will study the fluctuations of the matrix elements for
the desymmetrized quantum cat map and present a conjecture for the
distribution of the normalized matrix elements, namely that their
distribution is that of a certain weighted sum of traces of
independent matrices in SU(2). This is in contrast to generic chaotic
systems where the distribution is expected to be Gaussian. We will
show that the second and fourth moment of the distribution agree with
the conjecture, and also present some numerical evidence.
LAKSHMINARAYAN:
"Multifractal eigenstates of quantum chaos
and
the Thue-Morse sequence"
We show the emergence of automatic sequences such as the Thue-Morse
sequence and some of its generalizations in the eigenstates of the
quantum baker's map, a simple paradigmatic model of quantizing chaos.
This enables us to write nearly exactly a class of states in terms of
the spectra of such sequences, which turn out to be multifractal
measures. These states include those that are strongly scarred by
short classical periodic orbits, and their homoclinic excursions. Thus
the deterministic structural disorder of these quantum chaotic states
is seen to be captured by these ubiquitous, simple yet intriguing,
sequences.
While this happens for Hilbert space dimensions that are powers of
two, we touch upon the more general case where using an exactly
solvable permutation operator we "simplify" states of the quantum
baker's map. We point out that this operator is a crucial component of
both the art of shuffling cards, as well as the celebrated quantum
algorithm of Shor that factorizes large numbers with ease.
(1) "Multifractal eigenstates of quantum chaos and the Thue-Morse
sequence",
N. Meenakshisundaram, and AL. (
nlin.CD/0412042, to
appear in the Phys. Rev. E rapid comm..)
(2) "Shuffling cards, factoring numbers, and the quantum baker's map",
AL. (
nlin.CD/0505057).
LEBOEUF:
"Thermodynamics of a Fermi gas: average, fluctuations, universality"
The thermodynamic properties of Fermi gases show, on top of an average
behavior, characteristic fluctuations (as a function, for instance, of the
number of particles in the gas). These fluctuations may be described, in a
semiclassical theory, as sums over the periodic orbits of the
corresponding
classical potential. We present a general analysis of these fluctuations
and
describe how they depend on the regular or chaotic nature of the classical
dynamics. The typical size and correlations of the fluctuations may be
expressed, generically, in terms of Ruelle--Pollicott resonances.
The calculation of the level density of the gas is of particular interest.
We
show how this quantity is related to the problem of computing the number
of
partitions of an integer, and discuss the natural extensions (some of them
connected to random matrix theory) that emerge.
MAYER:
"Transfer operators and the automorphic functions for
modular groups"
I will review in this talk the relation between certain eigenfunctions of
the transfer
operators for the geodesic flows on modular surfaces, the so called period
functions, and
the holomorphic and real analytic automorphic functions for the
corresponding modular
groups.
NONNENMACHER:
"Open quantum bakers" (joint with M.Zworski)
We analyze the quantum version of a simple chaotic map on the torus (the
generalized baker's map), from the
semiclassical point of view. This system should serve as a model
for more realistic systems of quantum chaotic scattering (e.g. a 3-disk
scatterer on the plane).
The classical map has a fractal invariant hyperbolic set ("repeller"), and
invariant measures are
easy to construct explicitly. We are interested in the spectrum of the
quantum
"propagator", which should be identified with the resonances in the
scattering framework.
A first version of the quantized map provides a numerical
confirmation of the fractal Weyl law for the density
of quantum resonances (cf. M.Zworski's course). The exponent in that
law is related to the dimension of the classical repeller. We also discuss
the "profile function" (prefactor) of the density, and the eigenstates
associated
with the resonances.
Within a simplified quantization scheme, we manage to analytically compute
the full
resonance spectrum: the latter is shown to satisfy the fractal Weyl law.
Some
of the eigenstates have a simple fractal structure in phase space.
OLBRICH:
"Group cohomology and the Selberg zeta function"
(joint with U. Bunke)
We discuss the Selberg zeta function associated to the lift of the
geodesic flow to a vector bundle on the unit sphere bundle of
an infinite volume hyperbolic
manifold without cusps.
We are particularly interested in the understanding of its
zeroes and poles. Usually, one tries to establish a Selberg type trace
formula which leads to a description in terms of resonances
and topological data. Inspired by a conjecture of Patterson,
we propose an alternative description in terms of distributions supported
on the limit set which are covariant with respect to the action
of the fundamental group.
At least philosophically, this description is very
close to the transfer operator approach. In order to get
correct formulas for all singularities, one has to consider
not only covariant distributions on the limit set, but also
higher cohomology classes with coefficients in the space
of distributions supported on the limit set.
POLLICOTT: "Uniqueness of g-measures"
(joint with Anders Johansson and Anders Oberg)
g-measures are invariant probability measures m for a (one sided)
subshift of finite type T:X -> X for which the Jacobian
1/g := d(mT)/dm is continuous. These were introduced into ergodic
theory by Keane. We consider the classical problem of which functions g
have a unique m satisfying this identity (i.e., a unique g-measure).
In particular, we present a new approach to showing uniqueness under
the square summability condition of Johansson and Oberg which allows
us extend the results to a slightly broader setting.
RUDNICK: "Fluctuations of matrix elements and moments of zeta functions" (joint with K. Soundararajan)
The moments of the Riemann zeta function on the critical line have been an
object of study for several decades. Analogously one can study the moments
of
central values of families of L-functions.
Using results from Random Matrix Theory, Keating and Snaith have
recently given precise conjectures for the limiting values of moments
of central values of various families of L-functions.
We develop a simple method to establish lower bounds of the conjectured
order of magnitude for several such families of L-functions.
An application of these results is to a problem in "quantum chaos":
For many classically chaotic systems it is believed that the quantum
wave functions become uniformly distributed, that is the matrix
elements of smooth observables tend to the phase space average of the
observable ("Quantum Unique Ergodicity").
One then wants to describe the fluctuations of the
matrix elements around the phase space average. For generic systems,
it has been conjectured by Feingold and Peres that the matrix
coefficients have a limiting variance which decays at a certain rate
with a pre-factor which is given by the classical variance of the
observable, and that the value distribution is Gaussian.
We study this question for the modular domain. In this case,
for certain observables, the square of the matrix coefficients is
essentially the central value of an L-function (Watson's formula).
Luo and Sarnak have computed the variance and found that while the
rate of decay matches the Feingold-Peres predictions, the pre-factor
differs from the classical variance by an arithmetic quantity. we
study the the higher moments of the normalized matrix coefficients
and show that they blow up, therefore the fluctuations in this case
are not Gaussian.
ZAGIER: "Automorphic forms and their period functions"
ZELDITCH: "Complex zeros of real ergodic eigenfunctions"
Little is known about the distribution of real zeros of
real valued eigenfunctions of the Laplacian on a compact analytic
Riemannian manifold (M, g). If
Delta phi_{\lambda} = \lambda^2
\phi_{\lambda},
then the best result (Donnelly-Fefferman) is that
the (n-1) dimensional measure of Z_{\lambda} = {phi_{\lambda} = 0} is
bounded
above and below by a constant times lambda.
In our talk, we simplify the problem by
complexifying it. We holomorphically continue phi_{\lambda} to the
cotangent bundle
(equipped with its adapted complex structure) and study the limit
distribution
of its complex zeros. In the case where {\phi_{\lambda}} is a
sequence
of ergodic eigefunctions, the limit distribution of complex
zeros is given by the (1,1) form ddbar |\xi|_g. In particular, if the
geodesic flow of (M,g)
is ergodic, this limit formula is valid for 100% of the
eigenfunctions.