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Quantum chaos on graphs

Gregor Tanner

School of Mathematical Sciences,
University of Nottingham, University Park, Nottingham NG7 2RD, United Kingdom

Quantum graphs have recently been introduced as model systems to study quantum problems with chaotic classical limit (Kottos and Smilansky 1997). The most fascinating features of quantum graphs are that they can be constructed easily and almost at will covering a variety of classical limits such as chaotic dynamics, scattering or diffusive behaviour. Yet, the quantum mechanics can be formulated in terms of unitary propagators on finite Hilbert spaces. In addition, quantum graphs show many of the phenomena observed in more general quantum systems such as universality of the spectral statistics or Anderson localisation.

I will review recent developments on quantum graphs and generalise these concepts to quantum propagation on arbitrary, directed graphs (Tanner 2000). In its simplest version, the wave dynamics on the graph is solely determined by the topology of the graph given by the adjacency matrix, metric properties, that is, the length of the edges, and dynamical properties entering as (complex) transition amplitudes describing transitions between edges at the vertices. Necessary and sufficient conditions for a graph to be `quantisable' can be given (Pakonski et al 2002).

A specific quantum graph can in a natural way be associated with an ensemble of unitary matrices (Tanner 2001). The `classical' dynamics on the graph can be interpreted as a Markov chain defined on the graph with stochastic transition matrix $\bf T$ obtained from the unitary propagator on the graph (Kottos and Smilansky 1997, Pakonski et al 2001). I will formulate a conjecture linking universality of the statistical properties of the unitary matrix ensemble after ensemble average to the spectral gap of the stochastic transition matrix. More precisely, it is expected that the matrix ensemble follows random matrix statistics in the limit of large network size, if the spectral gap $\Delta$, that is, the minimal distance of eigenvalues of $\bf T$ from the unit circle, scales like (Tanner 2001)

\begin{displaymath}\lim_{N\to\infty} \frac{\log N}{ \Delta(N) N } = 0 \end{displaymath}

where $N$ is the number of edges in the graph. Some examples will be presented. Results on correlation functions for eigenvalues and spectral determinants will be discussed (Tanner 2002).

Kottos T and Smilansky U 1997 Phys. Rev. Lett. 79 4794
Pakonski P, $\dot{\rm Z}$yczkowski K and Kus M 2001 J. Phys. A 34 9393
Pakonski P, Tanner G and $\dot{\rm Z}$yczkowski K 2002 Line-graph families and their quantisation, preprint
Tanner G 2000 J. Phys. A 33 3567
Tanner G 2001 J. Phys. A 34 8485
Tanner G 2002 The autocorrelation function for spectral determinants of quantum graphs,
to appear in J. Phys. A (nlin.CD/0203027v2)

Classical and quantum spectra of intermittent dynamics

Gregor Tanner

School of Mathematical Sciences,
University of Nottingham, University Park, Nottingham NG7 2RD, United Kingdom

Intermittency is a typical phenomenon on the transition from order to chaos. The existence of intermittent behaviour in dynamical systems can in general be traced back to the presence of marginal stability either at a single periodic orbit or along the boundary of stable islands. This leads to nearly regular together with strongly chaotic dynamics in connected components of the phase space. Intermittency is typically accompanied by algebraic decay of correlation and long tail memory effects. In this talk, I will present techniques to calculate the spectra of Frobenius-Perron operators for intermittent systems in terms of periodic orbits. Implications for a periodic orbit quantisation of intermittent dynamics using Gutzwiller's trace formula will be discussed briefly.

Periodic orbits, which approach the marginal stable regime in phase space are characterised by a vanishing Lyapunov exponent

\lambda_p = \frac{\log\Lambda_p}{T_p} \to 0, \qquad \mbox{for}
\quad T_p \to \infty

with $\Lambda_p$ the largest eigenvalue of the Monodromy matrix along the orbit. The algebraic decay in the eigenvalues $\Lambda_p$ induces divergences in trace formula, which need to be removed systematically (in addition to the usual problem of overcoming convergence problems due to the exponential proliferation of periodic orbits). This leads to periodic orbit expansions of the trace of the Frobenius Perron operator in terms of families of periodic orbits converging towards the marginal stable region.

The talk is mainly based on the chapter on intermittency in the web-book by Cvitanovic et al. I will introduce the method for a specific 1d-piecewise linear map for which the algebraic decay behaviour can be calculated explicitly. Generalisations to arbitrary uni-model maps will be given. The stadium billiard will serve as an example to discuss modifications of the method for semiclassical periodic orbit formulas (Tanner 1997).

Related references
P. Cvitanovic et al Classical and Quantum Chaos - A Cyclist Treatise,
H. H. Rugh 1999 Invent. Math. 135 1
T. Prellberg 2002 Complete Determination of the Spectrum of a Transfer Operator associated with Intermittency, preprint (nlin.CD/0108044 v1)
G. Tanner, K. Hansen and J. Main 1996 Nonlinearity 9 1641
G. Tanner 1997 J. Phys. A. 30 2863
G. Tanner, K. Richter and J. M. Rost 2000 Review of Modern Physics 72 497

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