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OVERVIEW
A Research Training Network funded by the European Commission under the Human Potential Programme.
Any question? Send an e-mail to: info@quantumchaos.net
PROJECT OBJECTIVES.
It has been conjectured that, generically, the quantum energy levels of
individual, classically integrable systems are distributed like independent
random numbers, and that those of classically chaotic systems are distributed
like the eigenvalues of random matrices.
Likewise, the eigenfunctions of the Schrūdinger equation have been conjectured
to behave like Gaussian random functions in the semiclassical (short wavelength)
limit.
These and related conjectures have recently attracted intense interest
from both physicists and mathematicians. Physicists have used them to
develop theories for the behaviour of microelectronic devices that straddle
the border between quantum and classical mechanics. Mathematicians have
shown that they are connected to deep results in number theory (for example,
the zeros of the Riemann zeta function are also believed to be distributed
like the eigenvalues of random matrices), ergodic theory, and hyperbolic
geometry.
The goal of the research project on which the network will be based is
to prove results related to these ideas, and ultimately to prove the conjectures
themselves for particular families of simple model systems.
The models to be studied include simple maps, geodesic motion on surfaces
of constant negative curvature, graphs, and billiards.
The aim will be in each case to investigate the semiclassical limits of
the quantum eigenfunctions and eigenvalue distributions, and to determine
how these are related to the chaotic nature of the underlying classical
motion.
APPLICATIONS
The research programme is directed towards fundamental theoretical results.
However, at the currect rate of miniaturisation, microelectronic components
will enter the quantum regime in the next decade or two.
Examples of such devices (eg, quantum dots, resonant tunneling diodes,
etc) have already been constructed and are currently the subject of intense
experimental interest, both from the point of view of fundamental physics
and technological application.
The response characterstics (eg conductivity as a function of device shape,
and/or applied magnetic field) of such devices are modelled by treating
them as quantum chaotic systems.
Questions about correlations in quantum spectra and wavefunctions, and
their links with random matrix theory, are therefore directly related
to the key problem of understanding the (large) fluctuations in these
response characteristics.
TRAINING ASPECTS
In
the last several years, research in quantum chaos has undergone rapid
and unexpected development, while at the same time taking on a strongly
interdisciplinary character. By drawing together independent research
teams of shared interests but complementary specialities, the network
will provide young researchers in the field with the breadth of training
needed at the frontier.
Postdoctoral positions will be shared between individual members of different
teams, and will be built around collaborations on specific projects.
Researchers will learn at first-hand techniques from a range of disciplines
(number theory, hyperbolic geometry, ergodic theory, semiclassical asymptotics)
from leading practitioners.
The teams will also offer focussed postgraduate training, including advanced
courses and research supervision.
As part of its training function, the network will organise annual schools
in the areas of expertise of the teams.
The lecture series will be made available on the world-wide web as WebSeminars,
producing an enduring and accessible pedagogical archive for the field.