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30
The Riemann Zeros and Eigenvalue Asymptotics
 SIAM Rev
, 1999
"... 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 feat ..."
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Cited by 42 (5 self)
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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 randommatrix 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 RiemannSiegel 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.
How Chaotic is the Stadium Billiard? A Semiclassical Analysis
"... The impression gained from the literature published to date is that the spectrum of the stadium billiard can be adequately described, semiclassically, by the Gutzwiller periodic orbit trace formula together with a modified treatment of the marginally stable family of bouncing ball orbits. I show tha ..."
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Cited by 12 (0 self)
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The impression gained from the literature published to date is that the spectrum of the stadium billiard can be adequately described, semiclassically, by the Gutzwiller periodic orbit trace formula together with a modified treatment of the marginally stable family of bouncing ball orbits. I show that this belief is erroneous. The Gutzwiller trace formula is not applicable for the phase space dynamics near the bouncing ball orbits. Unstable periodic orbits close to the marginally stable family in phase space cannot be treated as isolated stationary phase points when approximating the trace of the Green function. Semiclassical contributions to the trace show an ¯h – dependent transition from hard chaos to integrable behavior for trajectories approaching the bouncing ball orbits. A whole region in phase space surrounding the marginal stable family acts, semiclassically, like a stable island with boundaries being explicitly ¯h–dependent. The localized bouncing ball states found in the billiard derive from this semiclassically stable island. The bouncing ball orbits themselves, however, do not contribute to individual eigenvalues in the spectrum. An EBK–like quantization of the regular bouncing ball eigenstates in the stadium can be derived. The stadium billiard is thus an ideal model for studying the influence of almost regular dynamics near marginally stable boundaries on quantum mechanics. This behavior is generically found at the border of classically stable islands in systems with a mixed phase space structure. PACS numbers: 05.45, 03.65.Sq 1
Transcritical bifurcations in nonintegrable Hamiltonian systems
, 2008
"... We report on transcritical bifurcations of periodic orbits in nonintegrable twodimensional 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. ..."
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We report on transcritical bifurcations of periodic orbits in nonintegrable twodimensional 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 timereversal 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 quantummechanical results. 1
Use of Harmonic Inversion Techniques in Semiclassical Quantization and Analysis of Quantum Spectra
, 1999
"... Harmonic inversion is introduced as a powerful tool for both the analysis of quantum spectra and semiclassical periodic orbit quantization. The method allows to circumvent the uncertainty principle of the conventional Fourier transform and to extract dynamical information from quantum spectra which ..."
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Cited by 2 (1 self)
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Harmonic inversion is introduced as a powerful tool for both the analysis of quantum spectra and semiclassical periodic orbit quantization. The method allows to circumvent the uncertainty principle of the conventional Fourier transform and to extract dynamical information from quantum spectra which has been unattainable before, such as bifurcations of orbits, the uncovering of hidden ghost orbits in complex phase space, and the direct observation of symmetry breaking effects. The method also solves the fundamental convergence problems in semiclassical periodic orbit theories  for both the BerryTabor formula and Gutzwiller's trace formula  and can therefore be applied as a novel technique for periodic orbit quantization, i.e., to calculate semiclassical eigenenergies from a finite set of classical periodic orbits. The advantage of periodic orbit quantization by harmonic inversion is the universality and wide applicability of the method, which will be demonstrated in this work for v...
FROM OPEN QUANTUM SYSTEMS TO OPEN QUANTUM MAPS
"... 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 relat ..."
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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.
Periodic Orbit Quantization of Chaotic Systems
"... 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 3disk scattering problem. The symmetry decompositions of the eigenspectra are the same ..."
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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 3disk 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...
Semiclassical Quantization and Analysis of Chaotic Systems
"... 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 ..."
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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 . . . .
Normal forms and complex periodic orbits in semiclassical expansions of Hamiltonian systems
"... 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 presentat ..."
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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 UPRESA 6083....
Effect of pitchfork bifurcations on the spectral statistics of Hamiltonian systems
, 2008
"... 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 sign ..."
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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 twofold: Beside the known effect of an enhanced periodic orbit contribution due to its peculiar ¯hdependence 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 socalled diagonal approximation. Our semiclassical predictions for both the coarsegrained density of states and the spectral rigidity, are in excellent agreement with corresponding quantummechanical results. 1 1
Periodic orbit theory for the HénonHeiles system in the continuum region
, 2004
"... We investigate the resonance spectrum of the HénonHeiles 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 ..."
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We investigate the resonance spectrum of the HénonHeiles 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 codimensiontwo bifurcation scenario, we are able to reproduce the coarsegrained quantummechanical density of states very accurately, including only a few stable and unstable orbits. 05.45.Mt,03.65.Sq I.