Dynamical Zeta Function Research Articles

Semiclassical analysis of energy level correlations for a disordered mesoscopic system.

A model for a mesoscopic system where hyperspherical impurities are placed at random in the interior of hypercubical billiard is studied. The disorder is characterized by the elastic mean free path $l$ that ranges from the ballistic to the diffusive regime. The energy level correlator $K(s)$ and the form factor $\stackrel{^}{K}(t)$ are calculated and characterized by analyzing the analytic structure of the dynamical zeta function of the corresponding classical system.

Functional equation for dynamical zeta functions of Milnor-Thurston type

A Milnor-Thurston type dynamical zeta functionζL(Z) is associated with a family of maps of the interval (−1, 1). Changing the direction of time produces a new zeta functionζ′L(Z). These zeta functions satisfy a functional equationζL(Z)ζ′L(eZ)=ζ0(Z) (where e amounts to sign changes and, generically,ζ≡01). The functional equation has non-trivial implications for the analytic properties ofζL(Z).

Meromorphic zeta functions for analytic flows

We extend to hyperbolic flows in all dimensions Rugh's results on the meromorphic continuation of dynamical zeta functions. In particular we show that the Ruelle zeta function of a negatively curved real analytic manifold extends to a meromorphic function on the complex plane.

Distribution of closed orbits for linear automorphisms of tori

In this paper we study the distribution properties of periodic orbits for the linear hyperbolic automorphisms of the d-torus. We first obtain an explicit expression of the dynamical zeta function and prove general equidistribution results similar to those obtained for axiom A flows. We then study in detail some families of periodic orbits living on invariant prime lattices: they have the property that the integral of any character along any single orbit can be reduced to a number theoretic exponential sum over a finite field. This fact enables us to obtain explicit estimates on their asymptotic distributional properties.

A family of rational zeta functions for the quadratic map

A family of dynamical zeta functions is introduced, that weigh each periodic orbit with integral powers of its multiplier. For the quadratic map, we prove that some of these functions are rational, and conjecture their general form.

Dynamical zeta functions for Artin's billiard and the Venkov-Zograf factorization formula

Dynamical zeta functions are expected to relate the Schrödinger operator's spectrum to the periodic orbits of the corresponding fully chaotic Hamiltonian system. The relationship is exact in the case of surfaces of constant negative curvature. The recently found factorization of the Selberg zeta function for the modular surface is known to correspond to a decomposition of the Schrödinger operator's eigenfunctions into two sets obeying different boundary condition on Artin's billiard. Here we express zeta functions for Artin's billiard in terms of generalized transfer operators, providing thereby a new dynamical proof of the above interpretation of the factorization formula. This dynamical proof is then extended to the Artin-Venkov-Zograf formula for finite coverings of the modular surface.

Dynamical Zeta Functions for Maps of the Interval

A dynamical zeta function $\zeta$ and a transfer operator $\mathcal {L}$ are associated with a piecewise monotone map $f$ of the interval [0, 1] and a weight function $g$. The analytic properties of $\zeta$ and the spectral properties of $\mathcal {L}$ are related by a theorem of Baladi and Keller under an assumption of "generating partition". It is shown here how to remove this assumption and, in particular, extend the theorem of Baladi and Keller to the case when $f$ has negative Schwarzian derivative.

A zeta function approach to the semiclassical quantization of maps

The quantum analogue of an area-preserving map on a compact phase-space is a unitary (evolution) operator which can be represented by a matrix of dimension L α ℏ −1. The semiclassical theory for the spectrum of the evolution operator will be reviewed with special emphasis on developing a dynamical zeta function approach, similar to the one introduced recently for the semiclassical quantization of hamiltonian flows.

Test of the periodic-orbit approximation in n-disk systems

The scattering of a point panicle in two dimensions from two (or three) equally-sized (and spaced) circular hard disks is one of the simplest classically hyperbolic scattering problems. Because of this simplicity such systems are well suited for the study of the semiclassical periodic-orbit approximation in the cycle expansion of the dynamical zeta function applied to a quantum-mechanical scattering problem. Especially the predictions of the semiclassical cycle expansion for the quantum-mechanical resonances can be tested in these n-disk systems. Whereas for high wave numbers the cycle expansion gives quite accurate results, there are systematic deviations for low wave numbers from the exact quantum-mechanical values. The low-lying quantum-mechanical resonance poles of the 2- and 3-disk problem are constructed and compared to the cycle-expansion results. The characteristic determinant of the scattering matrix is expanded in terms of simple traces which in turn are related to the classical periodic orbits and possible creeping contributions. It will be shown that for large separations of the disks the correct resonance-pole positions can be extracted just from the knowledge of the lowest traces whose semiclassical limit are the fundamental periodic orbits. Creeping-orbit corrections are shown to be small.

Semiclassical computations of energy levels

Different methods of semiclassical calculations of energy levels of two-dimensional ergodic models are discussed and compared. Special attention is given to the calculation of the dynamical zeta function via the Rieman-Siegel relations.

Resummation of classical and semiclassical periodic-orbit formulas.

The convergence properties of cycle expanded periodic orbit expressions for the spectra of classical and semiclassical time evolution operators have been studied for the open three disk billiard. We present evidence that both the classical and the semiclassical Selberg zeta function have poles. Applying a Pad\'{e} approximation on the expansions of the full Euler products, as well as on the individual dynamical zeta functions in the products, we calculate the leading poles and the zeros of the improved expansions with the first few poles removed. The removal of poles tends to change the simple linear exponential convergence of the Selberg zeta functions to an $\exp\{-n^{3/2}\}$ decay in the classical case and to an $\exp\{-n^2\}$ decay in the semiclassical case. The leading poles of the $j$th dynamical zeta function are found to equal the leading zeros of the $j+1$th one: However, in contrast to the zeros, which are all simple, the poles seem without exception to be {\em double}\/. The poles are therefore in general {\em not}\/ completely cancelled by zeros, which has earlier been suggested. The only complete cancellations occur in the classical Selberg zeta function between the poles (double) of the first and the zeros (squared) of the second dynamical zeta function. Furthermore, we find strong indications that poles are responsible for the presence of spurious zeros in periodic orbit quantized spectra and that these spectra can be greatly improved by removing the leading poles, e.g. by using the Pad\'{e} technique.

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.

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 [Formula: see text] 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 [Formula: see text], is emphasized.

Asymptotic distribution of the pseudo-orbits and the generalized Euler constant gamma Delta for a family of strongly chaotic systems.

Dynamical zeta functions, defined as Euler products over classical periodic orbits, have recently received enhanced attention as an important tool for the quantization of chaos. Their representation as a Dirichlet series over pseudo-orbits has proven to be particularly useful, since these series seem to possess in the general case much better convergence properties than the original Euler product. The convergence of the Dirichlet series depends crucially on the asymptotic distribution of the pseudo-orbits and thus on the ergodicity of the underlying dynamical system. It is shown that the lengths ${\mathit{l}}_{\mathit{n}}$ (or rather exp${\mathit{l}}_{\mathit{n}}$) of the classical periodic orbits play mathematically the role of generalized prime numbers. Based on the theory of Beurling's generalized prime numbers, we derive an exact law for the proliferation of psuedo-orbits for the Hadamard-Gutzwiller model, which is one of the main testing grounds of our ideas about quantum chaos. The strength of growth of the pseudo-orbits is determined by the ratio ZETA(2)/ZETA'(1), where ZETA(s) denotes the Selberg zeta function. Two explicit, complementary representations are given that allow the computation of this ratio solely from the length spectrum {${\mathit{l}}_{\mathit{n}}$} of the classical periodic orbits, or from the quantal energy spectrum {${\mathit{E}}_{\mathit{n}}$}. One of these relations depends exponentially on the generalized Euler constant ${\ensuremath{\gamma}}_{\mathrm{\ensuremath{\Delta}}}$, which is therefore also studied. The formulas are applied to two strongly chaotic systems. It turns out that our asymptotic law describes the mean proliferation of pseudo-orbits very well not only in the asymptotic region, but also surprisingly well down to the shortest pseudo-orbit.

Semiclassical quantization of multidimensional systems

The solution of a stationary k-dimensional Schrodinger equation H Psi =E Psi in the semiclassical limit h(cross) to 0 is reduced to a discrete (k-1)-dimensional quantum map psi '=T psi where the integral kernel (the matrix) T is built through classical trajectories corresponding to the classical Poincare map of the given problem. High-excited energy eigenvalues obey the quantization condition zeta s(E)=0 where the function zeta s(E)=det(1-T) coincides with the Selberg zeta function defined as the product over primitive periodic orbits. Different properties of the constructed Poincare map are discussed, in particular the Riemann-Siegel relation for the dynamical zeta function.

From classical periodic orbits to the quantization of chaos

We present a quantitative analysis of Selberg’s trace formula viewed as an exact version of Gutzwiller’s semiclassical periodic-orbit theory for the quantization of classically chaotic systems. Two main applications of the trace formula are discussed in detail, (i) The periodic-orbit sum rules giving a smoothing of the quantal energy-level density. (ii) The Selberg zeta function as a prototype of a dynamical zeta function defined as an Euler product over the classical periodic orbits and its analytic continuation across the entropy barrier by means of a Dirichlet series. It is shown how the long periodic orbits can be effectively taken into account by a universal remainder term which is explicitly given as an integral over an ‘orbit-selection function’. Numerical results are presented for the free motion of a point particle on compact Riemann surfaces (Hadamard-Gutzwiller model), which is the primary testing ground for our ideas relating quantum mechanics and classical mechanics in the case of strong chaos. Our results demonstrate clearly the crucial role played by the long periodic orbits. An exact rule for quantizing chaos is derived for such systems where the Dirichlet series representing the Selberg zeta function converges on the critical line. Explicit formulae are given for the computation of the abscissae of absolute and conditional convergence, respectively, of these dynamical Dirichlet series. For the two Riemann surfaces considered, it turns out that one can cross the entropy barrier, but that the critical line cannot be reached by a convergent Dirichlet series. It would seem that this is the main reason why the Riemann-Siegel lookalike formula, recently conjectured by M. V. Berry and J. P. Keating, fails in generating the lower-lying quantal energies for these strongly chaotic systems.

On dynamical zeta function.

The dynamical zeta function is usually defined as an infinite (and divergent) product over all primitive periodic orbits. It is possible to show that as variant Planck's over 2pi -->0 it can be represented as det(1-T), where the operator T(q,q') defines the semiclassical Poincare map. Here, certain consequences of this representation for chaotic systems are discussed. In particular, it is shown that the zeta function can be expressed through a subset of specially selected orbits, the error of this approximation being small as variant Planck's over 2pi -->0. Assuming that the chosen Poincare surface of section is divided into small cells of phase-space area of 2pi variant Planck's over 2pi, these trajectories are uniquely characterized by the requirement that they never go twice through the same cell.

Quantization of chaotic systems.

Starting from the semiclassical dynamical zeta function for chaotic Hamiltonian systems we use a combination of the cycle expansion method and a functional equation to obtain highly excited semiclassical eigenvalues. The power of this method is demonstrated for the anisotropic Kepler problem, a strongly chaotic system with good symbolic dynamics. An application of the transfer matrix approach of Bogomolny is presented leading to a significant reduction of the classical input and to comparable accuracy for the calculated eigenvalues.

Quantization of chaos.

We study a rule for quantizing chaos based on the dynamical zeta function defined by a Euler product over the classical periodic orbits as suggested by Gutzwiller's semiclassical trace formula. A test of our approximate quantization formula is carried out for the planar hyperbold billiard, which shows that at least the first 150 quantum energy levels can be generated.

A new zeta-function in Nielsen's theory and a universal product formula for dynamical zeta-functions

A new zeta-function in Nielsen's theory and a universal product formula for dynamical zeta-functions