Abstracts
Quantum Ergodicity - Basic Ideas and Recent Developments
Arnd Bäcker (Dresden)
Mini Course
Wavefunctions in chaotic quantum systems
In this course we give an introduction to
the behaviour of wavefunctions in chaotic systems.
Topics:
- classical billiards: examples, ergodicity, mixing, periodic orbits
- quantum billiards:
examples,
spectral staircase function,
eigenfunctions in integrable vs. chaotic systems,
mean behaviour of eigenfunctions,
statistical properties of wave functions (random wave model)
- quantum ergodicity:
semiclassical eigenfunction hypothesis,
quantum ergodicity theorem (+ a sketch of the mathematical
background),
simple examples,
quantum limits,
bouncing ball modes,
scars
rate of quantum ergodicity
- More recent topics:
autocorrelation function and quantum ergodicity,
quantum ergodicity for Poincaré Husimi functions, ...
Jens Bolte (Ulm)
Quantum ergodicity for matrix valued operators
We consider semiclassical pseudodifferential operators that are
(Weyl-) quantisations of matrix valued symbols such as, e.g.,
Dirac operators. In this setting to each eigenvalue of the principal
symbol of the Hamiltonian operator there corresponds an almost
invariant subspace of the quantum Hilbert space as well as a
classical dynamical system. The latter consists of a skew product
flow built over the Hamiltonian flow generated by the eigenvalue
of the principal symbol. Suitable invariant semiclassical
observables are identified and an Egorov theorem is proven for
them. Quantum ergodicity is then shown to hold for the projections
of eigenvectors to an almost invariant subspace provided the
corresponding skew product flow is ergodic.
Oliver Giraud (Orsay)
Periodic orbits and spectral statistics of pseudo-integrable systems
Pseudo-integrable systems classically have properties that are
intermediate between integrable and chaotic systems: their periodic
trajectories occur in pencils of periodic orbits, but singularities
can split these pencils, yielding an indetermination at the boundary
between the pencils. These mixed classical characteristics have their
quantum counterpart: the quantum energy spectrum displays
statistics which are intermediate between the statistics of integrable
and chaotic systems: the nearest-neighbour spacing distribution shows
level repulsion at the origin and exponential decrease at infinity; the
spectral form factor is intermediate between 0 (chaotic case) and 1
(integrable case) at infinity.
We will give several examples where it is possible to calculate
analytically these quantities.
Jon Harrison (Ulm)
Quantum spectral properties of graphs with spin
We compare the quantisations of compact mixing graphs with the Dirac and
Schrödinger operators. Terms in a diagramatic expansion of the spectral
form factors are related by averages over the group of
spin transformations. In random matrix theory the power series
expansions of the COE and CSE form factors are closely related. The same
relation is derived for quantum graphs with time-reversal symmetry
provided the spin transformations on the graph quantised with the Dirac
operator generate an irreducible quaternionic representation.
Stefan Heusler (Essen)
Semiclassical Foundation of Universality in Quantum Chaos
We present the semiclassical core of a proof of the so-called
Bohigas-Giannoni-Schmidt-conjecture: A dynamical sytem with full classical
chaos
has a quantum energy spectrum with universal spectral fluctuations on the
scale given
by the mean level spacing. We show how in the semiclassical limit all system
specific
properties fade away, leaving only ergodicity, combinatorics and topology in
action. We
thus build bridges between classical periodic orbits and the sigma model of
quantum field theory.
[1] Sebastian Müller, Stefan Heusler, Petr Braun, Fritz Haake and
Alexander Altland: Semiclassical Foundation of Universality in
Quantum Chaos,
nlin.CD/0401021
Tyll Krüger(Berlin)
The Shannon-McMillan theorem (AEP) for quantum sources and related topics
Pär Kurlberg (Gothenburg)
Mini Course
Introduction to quantum ergodicity and arithmetic quantum maps
The quantum mechanical counterpart to classical ergodicity is that
eigenfunctions of the quantized Hamiltonian (stationary states) should
be equidistributed in a certain sense. If a dynamical system is
classically ergodic, Schnirelman's theorem asserts that most
stationary states are equidistributed. However, subsequences of
exceptional eigenfunctions (scars) cannot be ruled out. On the other
hand, for some systems it seems reasonable to expect that no
exceptional eigenfunctions exist, so called Quantum Unique Ergodicity
(QUE). Other expected consequences of classical chaos is that the
value distribution of eigenfunctions, as well as the value
distribution of matrix coefficients of observables, should be
Gaussian.
We will discuss these topics in the context of cat maps, a simple
example of a chaotic dynamical system. Cat maps have strong
arithmetical properties, and as a result there can be very large
spectral degeneracies. In fact, when the spectral degeneracies are
huge, scarring actually occurs! On the other hand, spectral
degeneracies suggests that one should look for a family of commuting
operators. It turns out that such a family exists, and using this it
is possible to show that QUE holds for desymmetrized eigenfunctions.
Time permitting, we will make some comparisons with billiards on
modular surfaces, another system whose quantization has strong
arithmetical properties.
Karl-Olof Lindahl (Växjö)
Ergodicity in p-adic dynamics
We give necessary and sufficient conditions for
ergodicity of dynamics generated by p-adic power series. In [1]
it was shown that a p-adic monomial map of the form f(x)=xn
is minimal on all spheres of radius less than 1, centered at
x=1, if and only if n is a generator of the group of units
modulo p2. Moreover, in this case (in fact for all isometries
on compact open subsets of the p-adic numbers as shown in
[2]),
the properties of minimality, ergodicity, and unique ergodicity
are equivalent. These results were taken a bit further in [2], for
some special type of polynomials. We show that these results are
true also for a general power series near a fixed point [3].
References:
[1] M. Gundlach, A. Khrennikov, and K.-O. Lindahl. On ergodic
behavior of p-adic dynamical systems. Inf. Dimens. Anal.
Quantum
Prob. Relat. Top., 4(4):569-577, 2001.
[2] J. Bryk, C.E. Silva. p-adic measurable dynamical systems of
simple polynomials. Amer. Math. Monthly, accepted.
[2] K.-O. Lindahl. On conjugation, ergodicity and minimality of
p-adic holomorphic functions. Preprint, 2003.
Jens Marklof (Bristol)
Semiclassical limits of quantum maps with divided phase space - Weyl's law and quantum ergodicity
Holger Schanz (Göttingen)
Scars on graphs
A scar is a wavefunction with enhanced phase-space density in the
vicinity of an unstable periodic orbit. There is an established theory
of "weak scarring" (Heller et al.), where the enhanced probability
density is distributed over many states such that no particular state
deviates significantly from the ergodic limit. On the other hand, many
numerical experiments show "strong scars", where in one single state a
localization around a periodic orbit is clearly visible. A theory for
strong scarring in quantum graphs will be presented in the talk. It
shows that weak and strong scarring are completely unrelated
phenomena, at least for this particular model system.
[1] H. Schanz and T. Kottos: Scars on Quantum Networks Ignore the
Lyapunov Exponent,
Phys. Rev. Lett. 90 (2003) 234101.
Roman Schubert (Bristol)
Semi-classical behaviour of expectation values in Lagrangian states
for large times
We study the behaviour of quantum mechanical expectation values in
Lagrangian states in the limit \hbar \to 0 and t \to \infty.
We show that it
depends strongly on the dynamical properties of
the corresponding classical system. If the classical system is
strongly chaotic, i.e., Anosov, then the expectation values tend to a
universal limit. This can be viewed as an analogue of mixing in the
classical system. If the classical system is integrable, then
the expectation values need not converge, and if they converge
their limit depends on the initial state. An additional
difference occurs in the timescales for which we can prove this behaviour,
in the chaotic case we get only up to Ehrenfest time,
t \sim \ln(1/ \hbar), whereas for integrable system we have much
larger time range.
Workshop Home
Last modified: Jan 29 2004
Stefan Keppeler
(quantum.ergodicityATmatfys.lth.se)