22 Nov 1999
Thematic Workshop: Chaos - Classical and Quantum
15 Oct - 17 Dec 1999
Niels Bohr Institute - Blegdamsvej 17

Thursday, November 11, 1999
Quantum Chaos Seminar   Aud C, 15:15
Per Dahlquist,   Royal Institute of Technology, Stockholm

Title: Escape from intermittent repellers- Periodic orbit theory for crossover from exponential to algebraic decay


Abstract: We apply periodic orbit theory to study the asymptotic distribution of escape times from an open intermittent map. The survival probability can rigorously be bounded close to sums over periodic orbits. The dynamical zeta function exhibits a branch point which is associated with an asymptotic power law escape. By an analytic continuation technique we compute a pair of complex conjugate zeroes beyond the branch point, associated with a pre-asymptotic exponential decay. Applications to conductance fluctuations in quantum dots are discussed.

Thursday, November 18, 1999
Quantum Chaos Seminar   Aud C, 15:15
Stephen Creagh (with Niall Whelan),   University of Nottingham

Title: Chaotic tunnelling: statistics and anomaly


Abstract: In several dimensions, tunnelling rates across a potential barrier are quite complex in nature, appearing almost random as one moves from state to state. Certain features, however, can often be simply characterised by the underlying classical dynamics. Here we investigate statistical distributions calculated on the basis of dynamical characteristics of the complex classical orbit which crosses the barrier with minumum imaginary action. The calculation uses straightforward assumptions of random matrix theory and works quite well in generic problems. Often, however, there are strong deviations, and these seem to occur when the complex tunnelling orbit has a real extensioon which is periodic. It is proposed that the deviation is connected ed to recent work by Kaplan which relates deviant wavefunction statistics to scarring.

Tuesday, November 23, 1999 (NEW DATE)
Chaos Seminar   Aud C, 15:15
R. Artuso   U. dell'Insubria, Como

Title: Slow correlation decay


Abstract: We consider the problem of correlation decay for systems either non-uniformly hyperbolic (possessing marginally stable structures) or not hyperbolic at all (polygonal billiards). Numerical results, connections to transport and non-equilibrium properties, and theoretical open questions will be discussed.

Thursday, November 25, 1999
Quantum Chaos Seminar   Aud C, 16:30
Uzy Smilansky   Weizmann Institute, Israel

Title: Graphs, scattering, statistics and what not


Abstract: The universal "finger-prints" of classical chaos in quantum dynamics are observed in the spectral statistics in bound systems, and the cross-section (conductance) fluctuations in open systems. The same phenomena occur also in quantum graphs (networks), which, in spite of their simple construction, show the full complexity of fluctuations as in generic Hamiltonian systems. Thus, quantum graphs are an excellent paradigme for the study of the dynamical origins of the universal finger-prints mentioned above. The Schroedinger operator on graphs will be defined, its classical analogue will be shown to be chaotic, and the corresponding trace formula will be derived. The fluctuations will be analysed using these tools, and the relations to periodic orbit theory and combinatorics will be illustrated.

Tuesday, November 30, 1999
Quantum Chaos Seminar   Aud C, 15:15
Gabor Vattay   Eotvos University, Budapest

Title: Negative length orbits


Abstract: The Path-Length Spectra (PLS) of the reflection amplitudes in mesoscopic billiard systems attached to leads are studied. The PLS has peaks at the length of classical trajectories starting and ending at the entrance lead. These lengths are all positive numbers. Now we attach a superconductor to this system. On the normal-superconducor interface Andreev scattering occurs and electrons are retro-reflected as holes. The holes follow the trajectories of electrons backward in time. If in the billiard system diffractive scattering occurs the classical trajectories are not uniquely defined. The holes retracing the electron trajectories can follow paths different from the electron trajectory after diffractive scattering. This makes it possible to create diffractive periodic orbits with negative total actions, lengths and time periods. We present a system where these strange new creatures can be observed in the PLS.

Thursday, December 2, 1999
Quantum Chaos Seminar   Aud C, 15:15
Gregor Tanner   University of Nottingham

Title: Graphs, combinatorics and random matrix statistics


Abstract: Quantum graphs have recently been introduced as model systems to study the spectral statistics of linear wave problems with chaotic classical limits. I will generalise this approach by considering arbitrary, directed graphs with unitary transfer matrices. A special class of graphs, so--called binary graphs, is studied in more detail. For these, the conditions for periodic orbit pairs to be correlated (including correlations due to the unitarity of the transfer matrix) can be given explicitly. Using combinatorial techniques it is possible to obtain expressions for the form factor in terms of correlated periodic orbit pair contributions for some low--dimensional cases. Gradual convergence towards random matrix results is observed when increasing the number of vertices in the binary graphs.
Chaos Seminar   Aud C, 16:30
Marc Lefranc   Université de Lille

Title: Topological analysis of low-dimensional chaos: periodic orbits, knot theory, and symbolic dynamics


Abstract: It is by now well-known that unstable periodic orbits are invaluable tools to understand and master chaotic behavior. One promising approach is the template analysis proposed by Mindlin et al., and based on earlier work by Birman and Williams. Unstable periodic orbits in a 3D phase space are closed curves which can be characterized through invariants from knot theory. Two key properties are that: (i) the topological invariants of an UPO remain constant on the whole domain of existence of an UPO because of the uniqueness theorem, (ii) the topological organization of the UPO embedded in an attractor can be globally described by means of a branched 2D manifold, a ``template''. We will review the basic concepts of template analysis and show how it can be used to extract precise symbolic dynamical information from a set of UPO. By construction, this approach yields generating partitions which connect continuously to the one-dimensional and hyperbolic symbolic encodings.

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