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26
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 61 (10 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.
Symbolic dynamics and periodic orbits for the cardioid billiard
 J. PHYS. A
, 1997
"... : The periodic orbits of the strongly chaotic cardioid billiard are studied by introducing a binary symbolic dynamics. The corresponding partition is mapped to a topological wellordered symbol plane. In the symbol plane the pruning front is obtained from orbits running either into or through the c ..."
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Cited by 18 (8 self)
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: The periodic orbits of the strongly chaotic cardioid billiard are studied by introducing a binary symbolic dynamics. The corresponding partition is mapped to a topological wellordered symbol plane. In the symbol plane the pruning front is obtained from orbits running either into or through the cusp. We show that all periodic orbits correspond to maxima of the Lagrangian and give a complete list up to code length 15. The symmetry reduction is done on the level of the symbol sequences and the periodic orbits are classified using symmetry lines. We show that there exists an infinite number of families of periodic orbits accumulating in length and that all other families of geometrically short periodic orbits eventually get pruned. All these orbits are related to finite orbits starting and ending in the cusp. We obtain an analytical estimate of the KolmogorovSinai entropy and find good agreement with the numerically calculated value and the one obtained by averaging periodic orbits. F...
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 11 (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
F.: Pseudosymmetries of Anosov maps and Spectral statistics. Nonlinearity 13
, 2000
"... The statistics of the quantum eigenvalues of certain families of nonlinear maps on the twotorus are found not to belong to the universality classes one would expect from the symmetries of the (classical) dynamics the maps generate. These anomalies are shown to be caused by arithmetical quantum sym ..."
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Cited by 10 (0 self)
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The statistics of the quantum eigenvalues of certain families of nonlinear maps on the twotorus are found not to belong to the universality classes one would expect from the symmetries of the (classical) dynamics the maps generate. These anomalies are shown to be caused by arithmetical quantum symmetries which do not have a classical limit. They are related to the dynamics generated by associated linear torus maps on particular rational lattices that form the support of the quantum Wigner functions.
Quantum graphs with spin Hamiltonians, in this volume
, 2007
"... Abstract. The article surveys quantization schemes for metric graphs with spin. Typically quantum graphs are defined with the Laplace or Schrödinger operator which describe particles whose intrinsic angular momentum (spin) is zero. However, in many applications, for example modeling an electron (whi ..."
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Cited by 4 (1 self)
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Abstract. The article surveys quantization schemes for metric graphs with spin. Typically quantum graphs are defined with the Laplace or Schrödinger operator which describe particles whose intrinsic angular momentum (spin) is zero. However, in many applications, for example modeling an electron (which has spin1/2) on a network of thin wires, it is necessary to consider operators which allow spinorbit interaction. The article presents a review of quantization schemes for graphs with three such Hamiltonian operators, the Dirac, Pauli and Rashba Hamiltonians. Comparing results for the trace formula, spectral statistics and spinorbit localization on quantum graphs with spin Hamiltonians. 1.
Quantum Cat Maps With Spin 1/2
, 2000
"... We derive a semiclassical trace formula for quantized chaotic transformations of the torus coupled to a twospinor precessing in a magnetic eld. The trace formula is applied to semiclassical correlation densities of the quantum map, which, according to the conjecture of Bohigas, Giannoni and Schmit ..."
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Cited by 3 (0 self)
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We derive a semiclassical trace formula for quantized chaotic transformations of the torus coupled to a twospinor precessing in a magnetic eld. The trace formula is applied to semiclassical correlation densities of the quantum map, which, according to the conjecture of Bohigas, Giannoni and Schmit, are expected to converge to those of the circular symplectic ensemble (CSE) of random matrices. In particular, we show that the diagonal approximation of the spectral form factor for small arguments agrees with the CSE prediction. The results are confirmed by numerical investigations.
Geometrical Theory of Diffraction and Spectral Statistics
, 1999
"... We investigate the influence of diffraction on the statistics of energy levels in quantum systems with a chaotic classical limit. By applying the geometrical theory of diffraction we show that diffraction on singularities of the potential can lead to modifications in semiclassical approximations for ..."
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Cited by 2 (0 self)
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We investigate the influence of diffraction on the statistics of energy levels in quantum systems with a chaotic classical limit. By applying the geometrical theory of diffraction we show that diffraction on singularities of the potential can lead to modifications in semiclassical approximations for spectral statistics that persist in the semiclassical limit ~ # 0. This result is obtained by deriving a classical sum rule for trajectories that connect two points in coordinate space.
Semiclassical expansion of parametric correlation functions of the quantum time delay
, 2006
"... We derive semiclassical periodic orbit expansions for a correlation function of the Wigner time delay. We consider the Fourier transform of the twopoint correlation function, the form factor K(τ, x, y, M), that depends on the number of open channels M, a nonsymmetry breaking parameter x, and a sym ..."
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We derive semiclassical periodic orbit expansions for a correlation function of the Wigner time delay. We consider the Fourier transform of the twopoint correlation function, the form factor K(τ, x, y, M), that depends on the number of open channels M, a nonsymmetry breaking parameter x, and a symmetry breaking parameter y. Several terms in the Taylor expansion about τ = 0, which depend on all parameters, are shown to be identical to those obtained from Random Matrix Theory. PACS numbers: 03.65.Sq Semiclassical theories and applications. 05.45.Mt Semiclassical chaos (“quantum chaos”).
Spectral Statistics in Chaotic Systems With a Point Interaction
"... We consider quantum systems with a chaotic classical limit that are perturbed by a pointlike scatterer. The spectral form factor K() for these systems is evaluated semiclassically in terms of periodic and diffractive orbits. It is shown for order _2 and 93 that offdiagonal contributions to the f ..."
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We consider quantum systems with a chaotic classical limit that are perturbed by a pointlike scatterer. The spectral form factor K() for these systems is evaluated semiclassically in terms of periodic and diffractive orbits. It is shown for order _2 and 93 that offdiagonal contributions to the form factor which involve diffractive orbits cancel exactly the diagonal contributions from diffractive orbits, implying that the perturbation by the scatterer does not change the spectral statistic. We further show that parametric spectral statistics for these systems are universal for small changes of the strength of the scatterer.