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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.
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].
[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.

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