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.

We report on transcritical bifurcations of periodic orbits in non-integrable two-dimensional Hamiltonian systems. Different from all other bifurcations in Hamiltonian systems mentioned so far in the literature, a transcritical bifurcation cannot be described by any of the known generic normal forms. We derive the simplest normal form from the Poincaré map of the transcritical bifurcation and compare its analytical predictions against numerical results. We also study the stability of a transcritical bifurcation against perturbations of the Hamiltonian, and its unfoldings when it is destroyed by a perturbation. Although it does not belong to the known list of generic bifurcations, we show that it can exist in a system without any discrete spatial or time-reversal symmetry. Finally, we discuss the uniform approximation required to include transcritically bifurcating orbits in the semiclassical trace formula for the density of states of the quantum Hamiltonian and test it against fully quantum-mechanical results. 1

We demonstrate the utility of the periodic orbit description of chaotic motion by computing from a few periodic orbits highly accurate estimates of a large number of quantum resonances for the classically chaotic 3-disk scattering problem. The symmetry decompositions of the eigenspectra are the same for the classical and the quantum problem, and good agreement between the periodic orbit estimates and the exact quantum poles is observed. It is a characteristic feature of dynamical systems of few degrees of freedom that the motion is often organized around a few fundamental cycles. These short cycles capture the skeletal topology of the motion in the sense that any long orbit can approximately be pieced together from the fundamental cycles. Moreover, many quantities of interest can be computed as averages over periodic orbits. In ref. [1] a highly convergent expansion around short cycles has been introduced and applied to evaluation of classical chaotic averages. The goal of this lette...

Contents 1 Introduction 3 1.1 Motivation of semiclassical concepts . . . . . . . . . . . . . . . . . . . . . . 3 1.1.1 Basic semiclassical theories . . . . . . . . . . . . . . . . . . . . . . . 4 1.1.2 Convergence problems of the semiclassical trace formulae . . . . . . 5 1.2 Objective of this work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.2.1 High precision analysis of quantum spectra . . . . . . . . . . . . . . 7 1.2.2 Periodic orbit quantization . . . . . . . . . . . . . . . . . . . . . . . 9 1.3 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2 High precision analysis of quantum spectra 12 2.1 Circumventing the uncertainty principle . . . . . . . . . . . . . . . . . . . 13 2.2 Precision check of the periodic orbit theory . . . . . . . . . . . . . . . . . . 18 2.3 Ghost orbits and uniform semiclassical approximations . . . . . . . . . . . 22 2.3.1 The hyperbolic umbilic catastrophe . . . .

Bifurcations of periodic orbits as an external parameter is varied are a characteristic feature of generic Hamiltonian systems. Meyer's classification of normal forms provides a powerful tool to understand the structure of phase space dynamics in their neighborhood. We provide a pedestrian presentation of this classical theory and extend it by including systematically the periodic orbits lying in the complex plane on each side of the bifurcation. This allows for a more coherent and unified treatment of contributions of periodic orbits in semiclassical expansions. The contribution of complex fixed points is find to be exponentially small only for a particular type of bifurcation (the extremal one). In all other cases complex orbits give rise to corrections in powers of ~ and, unlike the former one, their contribution is hidden in the "shadow" of a real periodic orbit. 71 pages, 4 tables, 20 figures Unit'e de recherche de l'Universit'e Paris XI associ'ee au CNRS y CNRS UPRES-A 6083....

In this paper we show that for a class of open quantum systems satisfying a natural dynamical assumption (see §1.2) the study of the resolvent, and hence of scattering, and of resonances, can be reduced to the study of open quantum maps, that is of finite dimensional quantizations of canonical relations obtained by truncation of symplectomorphisms.

We present a quantitative semiclassical treatment of the effects of bifurcations on the spectral rigidity and the spectral form factor of the Hamiltonian quantum system defined by two coupled quartic oscillators, which on the classical level exhibits mixed phase space dynamics. We show that the signature of a pitchfork bifurcation is two-fold: Beside the known effect of an enhanced periodic orbit contribution due to its peculiar ¯h-dependence at the bifurcation, we demonstrate that the orbit pair born at the bifurcation gives rise to distinct deviations from universality slightly above the bifurcation. This requires a semiclassical treatment beyond the so-called diagonal approximation. Our semiclassical predictions for both the coarse-grained density of states and the spectral rigidity, are in excellent agreement with corresponding quantum-mechanical results. 1 1

We investigate the resonance spectrum of the Hénon-Heiles potential up to twice the barrier energy. The quantum spectrum is obtained by the method of complex coordinate rotation. We use periodic orbit theory to approximate the oscillating part of the resonance spectrum semiclassically and Strutinsky smoothing to obtain its smooth part. Although the system in this energy range is almost chaotic, it still contains stable periodic orbits. Using Gutzwiller’s trace formula, complemented by a uniform approximation for a codimension-two bifurcation scenario, we are able to reproduce the coarse-grained quantum-mechanical density of states very accurately, including only a few stable and unstable orbits. 05.45.Mt,03.65.Sq I.

In non-integrable Hamiltonian systems with mixed phase space and discrete symmetries, sequences of pitchfork bifurcations of periodic orbits pave the way from integrability to chaos. In extending the semiclassical trace formula for the spectral density, we develop a uniform approximation for the combined contribution of pitchfork bifurcation pairs. For a two-dimensional double-well potential and the familiar Hénon-Heiles potential, we obtain very good agreement with exact quantummechanical calculations. We also consider the integrable limit of the scenario which corresponds to the bifurcation of a torus from an isolated periodic orbit. For the separable version of the Hénon-Heiles system we give an analytical uniform trace formula, which also yields the correct harmonic-oscillator SU(2) limit at low energies, and obtain excellent agreement with the slightly coarse-grained quantum-mechanical density of states. 1

We propose a picture of Wigner function scars as a sequence of concentric rings along a twodimensional surface inside a periodic orbit. This is verified for a two-dimensional plane that contains a classical orbit of a Hamiltonian system with two degrees of freedom. The orbit is hyperbolic and the classical Hamiltonian is “softly chaotic ” at the energies considered. The stationary wave functions are the familiar mixture of scarred and random waves, but the spectral average of the Wigner functions in part of the plane is nearly that of a harmonic oscillator and individual states are also remarkably regular. These results are interpreted in terms of the semiclassical picture of chords and centres, which leads to a qualitative explanation of the interference effects that are manifest in the other region of the plane. The qualitative picture is robust with respect to a canonical transformation that bends the orbit plane. PACS numbers: 03.65.Sq, 05.45.+b Sixteen years have passed since Heller [1] detected scars of periodic orbit in individual eigenfunctions of chaotic systems. Explanations in terms of wave packets [1] or the semiclassical Green function [2,12] do predict an enhancement