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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.
Eigenvalue spacings for regular graphs
 IN IMA VOL. MATH. APPL
, 1999
"... We carry out a numerical study of fluctuations in the spectrum of regular graphs. Our experiments indicate that the level spacing distribution of a generic kregular graph approaches that of the Gaussian Orthogonal Ensemble of random matrix theory as we increase the number of vertices. A review of ..."
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Cited by 10 (5 self)
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We carry out a numerical study of fluctuations in the spectrum of regular graphs. Our experiments indicate that the level spacing distribution of a generic kregular graph approaches that of the Gaussian Orthogonal Ensemble of random matrix theory as we increase the number of vertices. A review of the basic facts on graphs and their spectra is included.
SUPERCOMPUTERS AND THE RIEMANN ZETA FUNCTION
"... The Riemann Hypothesis, which specifies the location of zeros ofthe Riemann zeta function, and thus describes the behavior of primes, is one of the most famous unsolved problems inmathematics, and extensive efforts have been made over more than a century to check it numerically for large sets of c ..."
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Cited by 6 (0 self)
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The Riemann Hypothesis, which specifies the location of zeros ofthe Riemann zeta function, and thus describes the behavior of primes, is one of the most famous unsolved problems inmathematics, and extensive efforts have been made over more than a century to check it numerically for large sets of cases. Recently a new algorithm, invented by the speaker and A. Schönhage, has been implemented, and used to compute over 175 million zeros near zero number 10^20. The new algorithm turned out to be over 5 orders of magnitude faster than older methods. The crucial ingredients in it are a rational function evaluation method similar to the GreengardRokhlin gravitational potential evaluation algorithm, the FFT, andbandlimited function interpolation. While the only present implementation is on a Cray, the algorithm can easily be parallelized.
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...
Harmonic inversion as a general method for periodic orbit quantization
 Nonlinearity II
, 1998
"... . In semiclassical theories for chaotic systems, such as Gutzwiller's periodic orbit theory, the energy eigenvalues and resonances are obtained as poles of a nonconvergent series g(w) = # n A n exp(is n w). We present a general method for the analytic continuation of such a nonconvergent serie ..."
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. In semiclassical theories for chaotic systems, such as Gutzwiller's periodic orbit theory, the energy eigenvalues and resonances are obtained as poles of a nonconvergent series g(w) = # n A n exp(is n w). We present a general method for the analytic continuation of such a nonconvergent series by harmonic inversion of the `time' signal, which is the Fourier transform of g(w). We demonstrate the general applicability and accuracy of the method on two different systems with completely different properties: the Riemann zeta function and the threedisk scattering system. The Riemann zeta function serves as a mathematical model for a bound system. We demonstrate that the method of harmonic inversion by filterdiagonalization yields several thousand zeros of the zeta function to about 12 digit precision as eigenvalues of small matrices. However, the method is not restricted to bound and ergodic systems, and does not require the knowledge of the mean staircase function, i.e. the Weyl t...
Some ideas on dynamical systems and the Riemann zeta function
"... Introduction In this note we explain how the theory of the Riemann zeta function naturally leads to the investigation of a class of dynamical systems on foliated spaces. The hope is that finding the right dynamical system will be an important step towards a better understanding of i(s). The entire a ..."
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Introduction In this note we explain how the theory of the Riemann zeta function naturally leads to the investigation of a class of dynamical systems on foliated spaces. The hope is that finding the right dynamical system will be an important step towards a better understanding of i(s). The entire approach carries over to motivic Lseries the most general kind of Lseries coming from arithmetic geometry. This is important for various reasons but for simplicity we will mostly be concerned with i(s). In the first section we recall some arguments from [D2] in favour of a possible cohomological interpretation of the Riemann zeta function. In the second section following [D3], [D4] we single out a class of foliated dynamical systems whose leafwise reduced cohomology has many of the formal properties desired in section one. We close with a number of further remarks and suggestions. For other approaches to<F
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 . . . .
Experiments on Acoustic Chaology And . . .
, 2001
"... We measure ultrasound transmission spectra and standing wave patterns of homogeneous and isotropic plates with free boundaries. Our work takes two directions. First, we study plates with the shape of known chaotic billiards, where we focus on issues of universality for spectral uctuations and level ..."
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We measure ultrasound transmission spectra and standing wave patterns of homogeneous and isotropic plates with free boundaries. Our work takes two directions. First, we study plates with the shape of known chaotic billiards, where we focus on issues of universality for spectral uctuations and level motion. We also study the effect of breaking the centerplane symmetry of the plate on spectral statistics and resonance widths, comparing our results to a random matrix model for approximate symmetries. This part of our work may be termed 'acoustic chaology', relating directly to issues studied in the field of quantum chaos. Second, we study a rectangular plate, where we focus on properties like mode conversion, acoustic bouncingball states, that are particular to these freely vibrating plates, but of general importance for elastodynamic systems. We seek to understand properties of these systems through a statistical approach and this part of our work may be termed 'statistical elastodynamics'...