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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 47 (9 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.
Spectral Statistics in the Quantized Cardioid Billiard
, 1994
"... : The spectral statistics in the strongly chaotic cardioid billiard are studied. The analysis is based on the first 11000 quantal energy levels for odd and even symmetry respectively. It is found that the levelspacing distribution is in good agreement with the GOE distribution of randommatrix the ..."
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Cited by 10 (7 self)
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: The spectral statistics in the strongly chaotic cardioid billiard are studied. The analysis is based on the first 11000 quantal energy levels for odd and even symmetry respectively. It is found that the levelspacing distribution is in good agreement with the GOE distribution of randommatrix theory. In case of the number variance and rigidity we observe agreement with the randommatrix model for shortrange correlations only, whereas for longrange correlations both statistics saturate in agreement with semiclassical expectations. Furthermore the conjecture that for classically chaotic systems the normalized mode fluctuations have a universal Gaussian distribution with unit variance is tested and found to be in very good agreement for both symmetry classes. By means of the Gutzwiller trace formula the trace of the cosinemodulated heat kernel is studied. Since the billiard boundary is focusing there are conjugate points giving rise to zeros at the locations of the periodic orbits in...
Spherical Pendulum, Actions, and Spin
, 1996
"... The classical and quantum mechanics of a spherical pendulum are worked out, including the dynamics of a suspending frame with moment of inertia `. The presence of two separatrices in the bifurcation diagram of the energymomentum mapping has its mathematical expression in the hyperelliptic nature o ..."
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Cited by 4 (4 self)
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The classical and quantum mechanics of a spherical pendulum are worked out, including the dynamics of a suspending frame with moment of inertia `. The presence of two separatrices in the bifurcation diagram of the energymomentum mapping has its mathematical expression in the hyperelliptic nature of the problem. Nevertheless, numerical computation allows to obtain the action variable representation of energy surfaces, and to derive frequencies and winding ratios from there. The quantum mechanics is also best understood in terms of these actions. The limit ` ! 0 is of particular interest, both classically and quantum mechanically, as it generates two copies of the frameless standard spherical pendulum. This is suggested as a classical interpretation of spin. 1 Introduction John Ross was born in the year when Schrodinger's equation and Born's statistical interpretation of the wave function were published. The triumph of quantum theory left only minor roles for classical mechanics, in...
PeriodicOrbit Theory of the Number Variance. . .
 Physica D
, 1994
"... We discuss the number variance \Sigma 2 (L) and the spectral form factor F (ø ) of the energy levels of bound quantum systems whose classical counterparts are strongly chaotic. Exact periodicorbit representations of \Sigma 2 (L) and F (ø ) are derived which explain the breakdown of universalit ..."
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Cited by 3 (2 self)
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We discuss the number variance \Sigma 2 (L) and the spectral form factor F (ø ) of the energy levels of bound quantum systems whose classical counterparts are strongly chaotic. Exact periodicorbit representations of \Sigma 2 (L) and F (ø ) are derived which explain the breakdown of universality, i. e., the deviations from the predictions of randommatrix theory. The relation of the exact spectral form factor F (ø ) to the commonly used approximation K(ø ) is clarified. As an illustration the periodicorbit representations are tested in the case of a strongly chaotic system at low and high energies including very longrange correlations up to L = 700. Good agreement between "experimental" data and theory is obtained. 1 Supported by Deutsche Forschungsgemeinschaft under Contract No. DFGSte 241/46 I Introduction Today it is commonly accepted that chaos in classical dynamics is a generic property of complex systems [1]. The most striking property of deterministic chaos is ...
Mode Fluctuations as Fingerprint of Chaotic and NonChaotic Systems
, 1996
"... : The modefluctuation distribution P (W ) is studied for chaotic as well as for nonchaotic quantum billiards. This statistic is discussed in the broader framework of the E(k; L) functions being the probability of finding k energy levels in a randomly chosen interval of length L, and the distribut ..."
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Cited by 3 (1 self)
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: The modefluctuation distribution P (W ) is studied for chaotic as well as for nonchaotic quantum billiards. This statistic is discussed in the broader framework of the E(k; L) functions being the probability of finding k energy levels in a randomly chosen interval of length L, and the distribution of n(L), where n(L) is the number of levels in such an interval, and their cumulants c k (L). It is demonstrated that the cumulants provide a possible measure for the distinction between chaotic and nonchaotic systems. The vanishing of the normalized cumulants C k , k 3, implies a Gaussian behaviour of P (W ), which is realized in the case of chaotic systems, whereas nonchaotic systems display nonvanishing values for these cumulants leading to a nonGaussian behaviour of P (W ). For some integrable systems there exist rigorous proofs of the nonGaussian behaviour which are also discussed. Our numerical results and the rigorous results for integrable systems suggest that a clear finge...
Multiplicities of periodic orbit lengths for nonarithmetic models
 J. Phys. A: Math. Gen
"... Multiplicities of periodic orbit lengths for nonarithmetic Hecke triangle groups are discussed. It is demonstrated both numerically and analytically that at least for certain groups the mean multiplicity of periodic orbits with exactly the same length increases exponentially with the length. The ma ..."
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Multiplicities of periodic orbit lengths for nonarithmetic Hecke triangle groups are discussed. It is demonstrated both numerically and analytically that at least for certain groups the mean multiplicity of periodic orbits with exactly the same length increases exponentially with the length. The main ingredient used is the construction of joint distribution of periodic orbits when group matrices are transformed by field isomorphisms. The method can be generalized to other groups for which traces of group matrices are integers of an algebraic field of finite degree. 1
Quantum and Arithmetical Chaos
, 2003
"... Summary. The lectures are centered around three selected topics of quantum chaos: the Selberg trace formula, the twopoint spectral correlation functions of Riemann zeta function zeros, and of the Laplace–Beltrami operator for the modular group. The lectures cover a wide range of quantum chaos appli ..."
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Summary. The lectures are centered around three selected topics of quantum chaos: the Selberg trace formula, the twopoint spectral correlation functions of Riemann zeta function zeros, and of the Laplace–Beltrami operator for the modular group. The lectures cover a wide range of quantum chaos applications and can serve as a nonformal introduction to mathematical methods of quantum chaos.
Fractal neurodyamics and quantum chaos : Resolving the mindbrain paradox through novel biophysics
 In E. Mac Cormac and M. Stamenov (Eds.) Fractals of brain, fractals of mind, Advances in Consciousness Research, 7 : John Page 24 Benjamin
, 1996
"... Abstract: A model of the mindbrain relationship is developed in which novel biophysical principles in brain function generate a dynamic possessing attributes consistent with consciousness and freewill. The model invokes a fractal link between neurodynamical chaos and quantum uncertainty. Transacti ..."
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Abstract: A model of the mindbrain relationship is developed in which novel biophysical principles in brain function generate a dynamic possessing attributes consistent with consciousness and freewill. The model invokes a fractal link between neurodynamical chaos and quantum uncertainty. Transactional wave collapse allows this link to be utilized predictively by the excitable cell, in a way which bypasses and complements formal computation. The formal unpredictability of the model allows mind to interact upon the brain, the predictivity of consciousness in survival strategies being selected as a trait by organismic evolution. 1 Mind and Brain, Chaos and Quantum Mechanics. 1.1 Paradigms in Scientific Discovery and The Enigma of Consciousness The twentieth century has seen the unification of the microscopic and cosmic realms of physics in theories such as inflation, in which symmetrybreaking of the fundamental forces is linked to cosmic expansion. Molecular biology has had equally epochmaking successes unravelling the intricate molecular mechanisms underlying living systems, from the genetic code through to developmental structures such as homeotic genes. Despite these conceptual advances, the principles by which the brain generates mind remain mysterious. The intractability of this central unresolved problem in science suggests its principles run deeper than the conventional biochemical description, requiring novel biophysical principles. This paper develops such a model based on linkage between the fractal aspect of chaotic neurodynamics
Nodal domains on quantum graphs
, 2003
"... Abstract. We consider the real eigenfunctions of the Schrödinger operator on graphs, and count their nodal domains. The number of nodal domains fluctuates within an interval whose size equals the number of bonds B. For well connected graphs, with incommensurate bond lengths, the distribution of the ..."
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Abstract. We consider the real eigenfunctions of the Schrödinger operator on graphs, and count their nodal domains. The number of nodal domains fluctuates within an interval whose size equals the number of bonds B. For well connected graphs, with incommensurate bond lengths, the distribution of the number of nodal domains in the interval mentioned above approaches a Gaussian distribution in the limit when the number of vertices is large. The approach to this limit is not simple, and we discuss it in detail. At the same time we define a random wave model for graphs, and compare the predictions of this model with analytic and numerical computations. 1. Introduction the Schrödinger operator on graphs The structure of the nodal set of wave functions reflects the type of the underlying classical flow. This was suspected and discussed a long time ago, [1, 2, 3, 4], and returned to the focus of current research once it was shown that not only the morphology, but the distribution of the number of nodal domains, is indicative of the nature of the