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48
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.
Random matrices and Lfunctions
 J. PHYS A MATH GEN
, 2003
"... In recent years there has been a growing interest in connections between the statistical properties of number theoretical Lfunctions and random matrix theory. We review the history of these connections, some of the major achievements and a number of applications. ..."
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Cited by 19 (7 self)
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In recent years there has been a growing interest in connections between the statistical properties of number theoretical Lfunctions and random matrix theory. We review the history of these connections, some of the major achievements and a number of applications.
Developments in random matrix theory
 J. Phys. A: Math. Gen
, 2000
"... In this preface to the Journal of Physics A, Special Edition on Random Matrix Theory, we give a review of the main historical developments of random matrix theory. A short summary of the papers that appear in this special edition is also given. 1 1 ..."
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Cited by 17 (0 self)
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In this preface to the Journal of Physics A, Special Edition on Random Matrix Theory, we give a review of the main historical developments of random matrix theory. A short summary of the papers that appear in this special edition is also given. 1 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 9 (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...
Correlations for pairs of closed geodesics
 Invent. Math
"... Abstract. In this article we consider natural counting problems for closed geodesics on negatively curved surfaces. We present asymptotic estimates for pairs of closed geodesics, the differences of whose lengths lie in a prescribed family of shrinking intervals. Related pair correlation problems hav ..."
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Cited by 9 (2 self)
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Abstract. In this article we consider natural counting problems for closed geodesics on negatively curved surfaces. We present asymptotic estimates for pairs of closed geodesics, the differences of whose lengths lie in a prescribed family of shrinking intervals. Related pair correlation problems have been studied in both Quantum Chaos and number theory. One of the most striking properties of negatively curved surfaces is the regularity of the distribution of the lengths of their closed geodesics. This is shown by the wellknown prime geodesic theorem. More precisely, let V denote a compact surface with a C ∞ Riemannian metric of strictly negative curvature. Given any closed geodesic
Spectral statistics for the Dirac operator on graphs
, 2002
"... We determine conditions for the quantisation of graphs using the Dirac operator for both two and four component spinors. According to the BohigasGiannoniSchmit conjecture for such systems with timereversal symmetry the energy level statistics are expected, in the semiclassical limit, to correspon ..."
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Cited by 8 (4 self)
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We determine conditions for the quantisation of graphs using the Dirac operator for both two and four component spinors. According to the BohigasGiannoniSchmit conjecture for such systems with timereversal symmetry the energy level statistics are expected, in the semiclassical limit, to correspond to those of random matrices from the Gaussian symplectic ensemble. This is confirmed by numerical investigation. The scattering matrix used to formulate the quantisation condition is found to be independent of the type of spinor. We derive an exact trace formula for the spectrum and use this to investigate the form factor in the diagonal approximation.
Some studies on arithmetical chaos in classical and quantum mechanics
 Internat. J. Modern Phys
, 1993
"... Several aspects of classical and quantum mechanics applied to a class of strongly chaotic systems are studied. The latter consists of single particles moving without external forces on surfaces of constant negative Gaussian curvature whose corresponding fundamental groups are supplied with an arithm ..."
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Cited by 7 (0 self)
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Several aspects of classical and quantum mechanics applied to a class of strongly chaotic systems are studied. The latter consists of single particles moving without external forces on surfaces of constant negative Gaussian curvature whose corresponding fundamental groups are supplied with an arithmetic structure. It is shown that the arithmetical features of the considered systems lead to exceptional properties of the corresponding spectra of lengths of closed geodesics (periodic orbits). The most significant one is an exponential growth of degeneracies in these geodesic length spectra. Furthermore, the arithmetical systems are distinguished by a structure that appears as a generalization of geometric symmetries. These pseudosymmetries occur in the quantization of the classical arithmetic systems as Hecke operators, which form an infinite algebra of selfadjoint operators commuting with the Hamiltonian. The statistical properties of quantum energies in the arithmetical systems have previously been identified as exceptional. They do not fit into the general scheme of random matrix theory. It is shown with the help of a simplified model for the spectral form factor
MATRIX ELEMENTS FOR THE QUANTUM CAT MAP: FLUCTUATIONS IN SHORT WINDOWS
, 2007
"... We study fluctuations of the matrix coefficients for the quantized cat map. We consider the sum of matrix coefficients corresponding to eigenstates whose eigenphases lie in a randomly chosen window, assuming that the length of the window shrinks with Planck’s constant. We show that if the length of ..."
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Cited by 7 (2 self)
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We study fluctuations of the matrix coefficients for the quantized cat map. We consider the sum of matrix coefficients corresponding to eigenstates whose eigenphases lie in a randomly chosen window, assuming that the length of the window shrinks with Planck’s constant. We show that if the length of the window is smaller than the square root of Planck’s constant, but larger than the separation between distinct eigenphases, then the variance of this sum is proportional to the length of the window, with a proportionality constant which coincides with the variance of the individual matrix elements corresponding to Hecke eigenfunctions.