R. Aurich

Statistical properties of highly excited quantum eigenstates of a strongly chaotic system

Abstract Statistical properties of highly excited quantal eigenstates are studied for the free motion (geodesic flow) on a compact surface of constant negative curvature (hyperbolic octagon) which represents a strongly chaotic system (K-system). The eigenstates are expanded in a circular-wave basis, and it turns out that the expansion coefficients behave as Gaussian pseudo-random numbers. It is shown that this property leads to a Gaussian amplitude distribution P (Ψ) in the semiclassical limit, i.e. the wave-functions behave as Gaussian random functions. This behaviour, which should hold for chaotic systems in general, is nicely confirmed for eigenstates lying 10 000 states above the ground state thus probing the semiclassical limit. In addition, the autocorrelation function and the path-correlation function are calculated and compared with a crude semiclassical Bessel-function approximation. Agreement with the semiclassical prediction is only found, if a local averaging is performed over roughly 1000 de Broglie wavelengths. On smaller scales, the eigenstates show much more structure than predicted by the first semiclassical approximation.

Periodic-orbit sum rules for the Hadamard-Gutzwiller model

Abstract It is shown how a variety of periodic-orbit sum rules can be used to extract information about a quantum mechanical system, whose classical counterpart is completely chaotic, from knowledge only of the classical system, and vice versa. The basis is the Selberg trace formula, an exact analogue for the Hadamard-Gutzwiller model of the semiclassical periodic-orbit theory of Gutzwiller, which relates the quantal energies to the lengths of the periodic orbits of the classical system. Statistical properties of the quantal energies in the low-energy region are studied, where we restrict ourselves to the level spacing and spectral rigidity.

On the periodic orbits of a strongly chaotic system

Abstract A point particle sliding freely on a two-dimensional surface of constant negative curvature (Hadamard-Gutzwiller model) exemplifies the simplest chaotic Hamiltonian system. Exploiting the close connection between hyperbolic geometry and the group SU(1,1)/⦅±1⦆, we construct an algorithm (symboliv dynamics), which generates the periodic orbits of the system. For the simplest compact Riemann surface having as its fundamental group the “octagon group”, we present an enumeration of more than 206 million periodic orbits. For the length of the n th primitive periodic orbit we find a simple expression in terms of algebraic numbers of the form m + √2 n ( m , n ϵN are governed by a particular Beatty sequence), which reveals a strange arithmetical structure of chaos. Knowledge of the length spectrum is crucial for quantization via the Selberg trace formula (periodic orbit theory), which in turn is expected to unravel the mystery of quantum chaos.

Energy-level statistics of the Hadamard-Gutzwiller ensemble

Abstract The statistical properties of the quantal energy levels of the Hadamard-Gutzwiller ensemble - whose classical members belong to the class of systems with hard chaos - are investigated. Based on a sample of 4500 energy levels, it is shown that the short-range statistics as nearest-neighbour spacing distributions are governed by the GOE predictions of random-matrix theory, which was first surmised by Wigner and by Landau and Smorodinsky for nuclear level statistics. This result strengthens the hypothesis that quantum systems with chaotic classical counterpart display level repulsion as predicted by random-matrix theory. However, the level statistics describing correlations over greater level distances deviate from the GOE predictions, which is explained as a simple consequrnce of the fact that the spectral rigidity Δ 3 ( L ) introduced by Dyson and Mehta saturates non-universally at a finite value Δ ∞ for L → Δ in complete agreement with the semiclassical theory developed by Berry.

Trace formulae for three-dimensional hyperbolic lattices and application to a strongly chaotic tetrahedral billiard

This paper is devoted to the quantum chaology of three-dimensional systems. A trace formula is derived for compact polyhedral billiards which tessellate the three-dimensional hyperbolic space of constant negative curvature. The exact trace formula is compared with Gutzwillers semiclassical periodic-orbit theory in three dimensions, and applied to a tetrahedral billiard being strongly chaotic. Geometric properties as well as the conjugacy classes of the defining group are discussed. The length spectrum and the quantal level spectrum are numerically computed allowing the evaluation of the trace formula as is demonstrated in the case of the spectral staircase N(E), which in turn is successfully applied in a quantization condition.Abstract This paper is devoted to the quantum chaology of three-dimensional systems. A trace formula is derived for compact polyhedral billiards which tessellate the three-dimensional hyperbolic space of constant negative curvature. The exact trace formula is compared with Gutzwillers semiclassical periodic-orbit theory in three dimensions, and applied to a tetrahedral billiard being strongly chaotic. Geometric properties as well as the conjugacy classes of the defining group are discussed. The length spectrum and the quantal level spectrum are numerically computed allowing the evaluation of the trace formula as is demonstrated in the case of the spectral staircase N ( E ), which in turn is successfully applied in a quantization condition.

Periodic orbits on the regular hyperbolic octagon

Abstract The length spectrum of closed geodesics on a compact Riemann surface corresponding to a regular octagon on the Poincare disc is investigated. The general form of the elements of the “octagon group”, a discrete subgroup of nu>SU(1,1) {± 1 } , in terms of 2 × 2 matrices is derived, and the previously conjectured law for the length of periodic orbits is proved analytically. An algorithm for the multiplicity of geodesics with a given length is developed, which leads to an efficient enumeration of the periodic orbits of this strongly chaotic system.

Exact theory for the quantum eigenstates of a strongly chaotic system

We present an exact theory for the quantum eigenstates of a strongly chaotic system, which consists of a point particle sliding freely on a two-dimensional compact surface of constant negative curvature. The main result is a general sum rule which relates an (almost) arbitrarily smoothed sum over the quantum wavefunctions to a sum over the lengths of the classical orbits. As an example, we apply this formula to the Gaussian smoothed sum over wavefunctions. The question of “scars” as imprints of individual closed orbits is discussed, and the connection with Bogomolnys semiclassical theory is explicitly worked out. The investigation of the amplitude distribution P(Ψ) reveals that the higher excited wavefunctions Ψ behave as Gaussian random waves.

Periodic-orbit theory of the number variance Σ2(L) of strongly chaotic systems

Abstract We discuss the number variance Σ 2 ( L ) and the spectral form factor F ( τ ) of the energy levels of bound quantum systems whose classical counterparts are strongly chaotic. Exact periodic-orbit representations of Σ 2 ( L ) and F ( τ ) are derived which explain the breakdown of universality, i.e., the deviations from the predictions of random-matrix theory. The relation of the exact spectral form factor F ( τ ) to the commonly used approximation K ( τ ) is clarified. As an illustration the periodic-orbit representations are tested in the case of a strongly chaotic system at low and high energies including very long-range correlations up to L = 700. Good agreement between “experimental” data and theory is obtained.

QUANTIZATION RULES FOR STRONGLY CHAOTIC SYSTEMS

We discuss the quantization of strongly chaotic systems and apply several quantization rules to a model system given by the unconstrained motion of a particle on a compact surface of constant negative Gaussian curvature. We study the periodic-orbit theory for distinct symmetry classes corresponding to a parity operation which is always present when such a surface has genus two. Recently, several quantization rules based on periodic orbit theory have been introduced. We compare quantizations using the dynamical zeta function Z(s) with the quantization condition where a periodic-orbit expression for the spectral staircase N(E) is used. A general discussion of the efficiency of periodic-orbit quantization then allows us to compare the different methods. The system dependence of the efficiency, which is determined by the topological entropy τ and the mean level density , is emphasized.

Crossing the entropy barrier of dynamical zeta functions

Dynamical zeta functions are an important tool to quantize chaotic dynamical systems. The basic quantization rules require the computation of the zeta functions on the real energy axis, where their Euler product representations running over the classical periodic orbits usually do not converge due to the existence of the so-called entropy barrier determined by the topological entropy of the classical system. We show that the convergence properties of the dynamical zeta functions rewritten as Dirichlet series are governed not only by the well-known topological and metric entropy, but depend crucially on subtle statistical properties of the Maslov indices and of the multiplicities of the periodic orbits that are measured by a new parameter for which we introduce the notion of a third entropy. If and only if the third entropy is nonvanishing, one can cross the entropy barrier; if it exceeds a certain value, one can even compute the zeta function in the physical region by means of a convergent Dirichlet series. A simple statistical model is presented which allows to compute the third entropy. Four examples of chaotic systems are studied in detail to test the model numerically.