Arnd Bäcker (Dresden)

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)

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)

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)

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)

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:

Tyll Krüger(Berlin)

Pär Kurlberg (Gothenburg)

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)

We give necessary and sufficient conditions for ergodicity of dynamics generated by

References:

[1] M. Gundlach, A. Khrennikov, and K.-O. Lindahl. On ergodic behavior of

[2] J. Bryk, C.E. Silva.

[2] K.-O. Lindahl. On conjugation, ergodicity and minimality of

Jens Marklof (Bristol)

Holger Schanz (Göttingen)

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:

Roman Schubert (Bristol)

We study the behaviour of quantum mechanical expectation values in Lagrangian states in the limit

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Last modified: Jan 29 2004 Stefan Keppeler (