Date: Mon, 18 Sep 1995 17:10:10 +0200
From: ceb@ihp.jussieu.fr
Semestre de physique au centre Emile Borel (IHP)
15 Septembre 95- 14 Fevrier 96
CHAOS ET QUANTIFICATION
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Resume du cours de Frank Steiner:
" Quantum chaos and the Selberg trace formula "
1er cours le lundi 2 octobre a l'IHP, amphi Darboux, de 14h30 a 16h.
Re'sume':
The course will consist of 3 one and a half hour lectures. I shall discuss a
prototype example of both classical and quantum chaos, i.e. , the free motion
of a particle (geodesic flow) on two- and three-dimensional manifolds of
constant negative Gaussian curvature. The quantization of the dynamics will
be carried out via the Selberg trace formula. Analytical as well as numerical
applications of the trace formula will be discussed.
Topics to be covered :
- Classical motion on hyperbolic manifolds (Hadamard, Artin,...), and the
length spectrum of periodic orbits. Huber's law, prime geodesic theorem.
- The eigenvalue problem of the Laplace-Beltrami operator on hyperbolic
manifolds
- The Selberg trace formula (for the last two topics see also the course of
Venkov)
- The Selberg and related Ruelle zeta function; their analytic properties;
functional equation; analogue of the Riemann hypothesis (for the last topic
see also the course of Rudnick)
- The trace of the heat kernel and an exact Weyl formula
- The MP zeta function, and zeta function regularization of the spectral
determinant; the generalized Euler constant
- Crossing the entropy barrier; computation of the non-trivial zeros of
Selberg and Riemann zeta
- Connection with Beurling's theory of generalized prime numbers
- Inverse quantum chaology ("Can one hear the periodic orbits .....")
- Discussion of various quantization rules
- Local and global energy-level-statistics; periodic orbit theory for the
number variance; universal signatures of quantum chaos
- Gaussian random behaviour of chaotic wave functions in the semiclassical
limit; are there scars ?