Comparison between formulae for the counting functions of the heights t n of the Riemann zeros and of semiclassical quantum eigenvalues En suggests that the t n are eigenvalues of an (unknown) hermitean operator H, obtained by quantizing a classical dynamical system with hamiltonian H cl . Many features of H cl are provided by the analogy; for example, the "Riemann dynamics" should be chaotic and have periodic orbits whose periods are multiples of logarithms of prime numbers. Statistics of the t n have a similar structure to those of the semiclassical En ; in particular, they display random-matrix universality at short range, and nonuniversal behaviour over longer ranges. Very refined features of the statistics of the t n can be computed accurately from formulae with quantum analogues. The Riemann-Siegel formula for the zeta function is described in detail. Its interpretation as a relation between long and short periodic orbits gives further insights into the quantum spectral fluctuations. We speculate that the Riemann dynamics is related to the trajectories generated by the classical hamiltonian H cl = XP. Key words. spectral asymptotics, number theory AMS subject classifications. 11M26, 11M06, 35P20, 35Q40, 41A60, 81Q10, 81Q50 PII. S0036144598347497 1.

: The periodic orbits of the strongly chaotic cardioid billiard are studied by introducing a binary symbolic dynamics. The corresponding partition is mapped to a topological well-ordered symbol plane. In the symbol plane the pruning front is obtained from orbits running either into or through the cusp. We show that all periodic orbits correspond to maxima of the Lagrangian and give a complete list up to code length 15. The symmetry reduction is done on the level of the symbol sequences and the periodic orbits are classified using symmetry lines. We show that there exists an infinite number of families of periodic orbits accumulating in length and that all other families of geometrically short periodic orbits eventually get pruned. All these orbits are related to finite orbits starting and ending in the cusp. We obtain an analytical estimate of the Kolmogorov-Sinai entropy and find good agreement with the numerically calculated value and the one obtained by averaging periodic orbits. F...

For relativistic particles with spin 1/2, which are described by the Dirac equation, a semiclassical trace formula is introduced that incorporates expectation values of observables in eigenstates of the Dirac-Hamiltonian. Furthermore, the semiclassical limit of an average of expectation values is expressed in terms of a classical average of the corresponding classical observable. 1 E-mail address: bol@physik.uni-ulm.de 1 Introduction The three decades that passed since the completion of Gutzwiller's pioneering work on the trace formula [1] (see also [2]) have witnessed a flourishing development of semiclassical methods in connection with trace formulae. Gutzwiller himself, e.g., applied his semiclassical representation of the spectral density in terms of a sum over classical periodic orbits to the quantisation of the anisotropic Kepler problem [1, 3]. Later, in the context of the then developing field of quantum chaos, it was realised that the Gutzwiller trace formula can quite gen...

We derive a semiclassical trace formula for quantized chaotic transformations of the torus coupled to a two-spinor precessing in a magnetic eld. The trace formula is applied to semiclassical correlation densities of the quantum map, which, according to the conjecture of Bohigas, Giannoni and Schmit, are expected to converge to those of the circular symplectic ensemble (CSE) of random matrices. In particular, we show that the diagonal approximation of the spectral form factor for small arguments agrees with the CSE prediction. The results are confirmed by numerical investigations.

. We determine the asymptotics of the two-point correlation function for quantum systems with half-integer spin which show chaotic behaviour in the classical limit using a method introduced by Bogomolny and Keating [Phys. Rev. Lett. 77 (1996) 1472--1475]. For time-reversal invariant systems we obtain the leading terms of the two-point correlation function of the Gaussian symplectic ensemble. Special attention has to be paid to the role of Kramers' degeneracy. PACS numbers: 03.65.Sq, 05.45.Mt z E-mail address: kep@physik.uni-ulm.de Two-point correlations of the GSE from periodic orbits 2 Understanding correlations of energy levels of quantum mechanical systems whose classical limit exhibits chaotic motion is one of the major topics in quantum chaos. The bridge between quantum mechanics and classical mechanics is provided by the Gutzwiller trace formula [1] which relates the quantum mechanical density of states d(E) = P n ffi(E \Gamma E n ) to a sum over periodic orbits of the cor...

By continuation from the hyperbolic limit of the cardioid billiard we show that there isan abundance of bifurcationsin the family of limaçon billiards. The statistics of these bifurcation shows that the size of the stable intervals decreases with approximately the same rate as their number increases with the period. In particular, we give numerical evidence that arbitrarily close to the cardioid there are elliptic islands due to orbits created in saddle node bifurcations. This shows explicitly that if in this one parameter family of mapsergodicity occursfor more than one parameter the set of these parameter values hasa complicated structure.

We consider quantum systems with a chaotic classical limit that are perturbed by a point-like scatterer. The spectral form factor K(-) for these systems is evaluated semiclassically in terms of periodic and diffractive orbits. It is shown for order _2 and 9-3 that off-diagonal contributions to the form factor which involve diffractive orbits cancel exactly the diagonal contributions from diffractive orbits, implying that the perturbation by the scatterer does not change the spectral statistic. We further show that parametric spectral statistics for these systems are universal for small changes of the strength of the scatterer.

We investigate the influence of diffraction on the statistics of energy levels in quantum systems with a chaotic classical limit. By applying the geometrical theory of diffraction we show that diffraction on singularities of the potential can lead to modifications in semiclassical approximations for spectral statistics that persist in the semiclassical limit ~ # 0. This result is obtained by deriving a classical sum rule for trajectories that connect two points in coordinate space.

We consider quantum systems with a chaotic classical limit that depend on an external parameter, and study correlations between the spectra at different parameter values. In particular, we consider the parametric spectral form factor K(τ, x) which depends on a scaled parameter difference x. For parameter variations that do not change the symmetry of the system we show by using semiclassical periodic orbit expansions that the small τ expansion of the form factor agrees with Random Matrix Theory for systems with and without time reversal symmetry. PACS numbers: 03.65.Sq Semiclassical theories and applications. 05.45.Mt Semiclassical chaos (“quantum chaos”).

Introduction A long standing problem in quantum chaology is the precise formulation of the conjecture of Bohigas, Giannoni and Schmit (BGS) [1] which states that, generically, the quantum spectral statistics of systems whose classical analogue is chaotic can be described by the average statistical behaviour of the eigenvalues of large random matrices. In contrast, Berry and Tabor [2] conjectured that for integrable systems the statistical distribution of eigenvalues is that of a Poisson process. Whereas in the latter case integrability is precisely dened by the Liouville-Arnold theorem [3], there is no general consensus as to what should be sucient conditions for the BGS conjecture to hold. It is generally believed that hyperbolicity, i. e. positive Lyapunov exponents, is sucient for observing statistics as in random matrix theory (RMT). However, since this is a very strong property, one might ask whether weaker conditions would suce. Actually, this question has already been