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224
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 52 (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.
Geometric Properties Of Eigenfunctions
 Russian Math. Surveys
"... We give an overview of some new and old results on geometric properties of eigenfunctions of Laplacians on Riemannian manifolds. We discuss the properties of nodal sets and critical points, the number of nodal domains, as well as asymptotic properties of eigenfunctions in the high energy limit ( ..."
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Cited by 25 (3 self)
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We give an overview of some new and old results on geometric properties of eigenfunctions of Laplacians on Riemannian manifolds. We discuss the properties of nodal sets and critical points, the number of nodal domains, as well as asymptotic properties of eigenfunctions in the high energy limit (such as weak* limits, the rate of growth of L p norms, and the relationship between positive and negative parts of eigenfunctions). 1. introduction It is wellknown that on a compact Riemannian manifold M one can choose an orthonormal basis of L 2 (M) consisting of eigenfunctions ' j of satisfying ' j + j ' j = 0; (1) where 0 = 0 < 1 2 : : : are the eigenvalues. The purpose of this survey paper is to present some recent (and not so recent) results about the asymptotics of Laplace eigenfunctions on compact manifolds. We focus here mainly on results about the nodal sets, asymptotic L p bounds and the problem of determining weaklimits of expected values (i.e. quantum ...
Scattering on Compact Manifolds With Infinitely Thin Horns
"... In the paper [25]... In this paper we consider the quantum mechanical scattering in a hedgehogshaped space which is constructed by gluing a finite number of halflines to distinct points of a compact Riemannian manifold of dimension less than four. The Hamiltonian of a quantum particle in such a sy ..."
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Cited by 21 (5 self)
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In the paper [25]... In this paper we consider the quantum mechanical scattering in a hedgehogshaped space which is constructed by gluing a finite number of halflines to distinct points of a compact Riemannian manifold of dimension less than four. The Hamiltonian of a quantum particle in such a system coincides with a Schrödinger operator on the punctured manifold (the points of gluing are removed) and with the free Schrödinger operator on each halfline. At the gluing points, some boundary conditions are imposed. In particular, the Schröodinger operator in a magnetic field is included in our scheme. The approach we use is based on the Krein resolvent formula from operator extension theory [50], therefore in Sec. 1 we give a very brief sketch of results needed from this theory. Sec. 2 is devoted to the construction of Schrödinger operators on the hedgehogshaped space; we use the theory of boundary value spaces [35] to describe all possible kinds of boundary conditions defining the Schrödinger operators. We distinguish among them operators of "Dirichlet" and of "Neumann" type. It is worth noting that the results of Sec. 2 are valid for all Riemannian manifolds of dimension less than four, not only for the compact ones. In principle, the definition of the Schrödinger operator on a hedgehogshaped space may be given in the framework of pseudodifferential operator theory on such a space [66], but our approach is more convenient for investigating the scattering parameters and connected with the approach to spectral problems for point perturbations on Riemannian manifolds [8], [9]...
Semiclassical trace formulae and eigenvalue statistics in quantum chaos Open Sys
 Information Dyn
, 1999
"... A detailed discussion of semiclassical trace formulae is presented and it is demonstrated how a regularized trace formula can be derived while dealing only with finite and convergent expressions. Furthermore, several applications of trace formula techniques to quantum chaos are reviewed. Then local ..."
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Cited by 17 (4 self)
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A detailed discussion of semiclassical trace formulae is presented and it is demonstrated how a regularized trace formula can be derived while dealing only with finite and convergent expressions. Furthermore, several applications of trace formula techniques to quantum chaos are reviewed. Then local spectral statistics, measuring correlations among finitely many eigenvalues, are reviewed and a detailed semiclassical analysis of the number variance is given. Thereafter the transition to global spectral statistics, taking correlations among infinitely many quantum energies into account, is discussed. It is emphasized that the resulting limit distributions depend on the way one passes to the global scale. A conjecture on the distribution of the fluctuations of the spectral staircase is explained in this general context and evidence supporting the conjecture is discussed. 1 Lectures held at the 3rd International Summer School/Conference Let’s face chaos through nonlinear dynamics at
Action Integrals and Energy Surfaces of the Kovalevskaya Top
 Bifurcation and Chaos
, 1997
"... The different types of energy surfaces are identified for the Kovalevskaya problem of rigid body dynamics, on the basis of a bifurcation analysis of Poincar'e surfaces of section. The organization of their foliation by invariant tori is qualitatively described in terms of Poincar'eFomenko ..."
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Cited by 14 (9 self)
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The different types of energy surfaces are identified for the Kovalevskaya problem of rigid body dynamics, on the basis of a bifurcation analysis of Poincar'e surfaces of section. The organization of their foliation by invariant tori is qualitatively described in terms of Poincar'eFomenko stacks. The individual tori are then analysed for sets of independent closed paths, using a new algorithm based on Arnold's proof of the Liouville theorem. Once these paths are found, the action integrals can be calculated. Energy surfaces are constructed in the space of action variables, for six characteristic values of energy. The data are presented in terms of color graphs that give an intuitive access to this highly complex integrable system. to be submitted to: International Journal of Bifurcation and Chaos 1 1. Introduction Among the integrable systems of classical mechanics, the Kovalevskaya case of rigid body dynamics has proved to be one of the most fascinating in its combination of mathe...
Elliptic Islands Appearing in NearErgodic Flows
, 1998
"... It is proved that periodic and homoclinic trajectories which are tangent to the boundary of any scattering (ergodic) billiard produce elliptic islands in the `nearby' Hamiltonian flows; i.e., in a family of twodegreesoffreedom smooth Hamiltonian flows which converge to the singular billia ..."
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Cited by 13 (6 self)
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It is proved that periodic and homoclinic trajectories which are tangent to the boundary of any scattering (ergodic) billiard produce elliptic islands in the `nearby' Hamiltonian flows; i.e., in a family of twodegreesoffreedom smooth Hamiltonian flows which converge to the singular billiard flow smoothly where the billiard flow is smooth and continuously where it is continuous. Such Hamiltonians exist; Indeed, sufficient conditions are supplied, and thus it is proved that a large class of smooth Hamiltonians converges to billiard flows in this manner. These results imply that ergodicity may be lost in the physical setting, where smooth Hamiltonians which are arbitrarily close to the ergodic billiards, arise.
Quantum Mechanics and Semiclassics of Hyperbolic nDisk Scattering Systems
 Physics Reports 309
, 1999
"... The scattering problems of a scalar point particle from an assembly of 1 < n < ∞ nonoverlapping and disconnected hard disks, fixed in the twodimensional plane, belong to the simplest realizations of classically hyperbolic scattering systems. Their simplicity allows for a detailed study of the ..."
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Cited by 12 (1 self)
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The scattering problems of a scalar point particle from an assembly of 1 < n < ∞ nonoverlapping and disconnected hard disks, fixed in the twodimensional plane, belong to the simplest realizations of classically hyperbolic scattering systems. Their simplicity allows for a detailed study of the quantum mechanics, semiclassics and classics of the scattering. Here, we investigate the connection between the spectral properties of the quantummechanical scattering matrix and its semiclassical equivalent based on the semiclassical zetafunction of Gutzwiller and Voros. We construct the scattering matrix and its determinant for any nonoverlapping ndisk system (with n < ∞) and rewrite the determinant in such a way that it separates into the product over n determinants of 1disk scattering matrices – representing the incoherent part of the scattering from the ndisk system – and the ratio of two mutually complex conjugate determinants of the genuine multiscattering matrix M which is of KorringaKohnRostokertype and which represents the coherent multidisk aspect of the ndisk scattering. Our quantummechanical calculation is welldefined at every step, as the onshell T–matrix and the multiscattering kernel M−1 are shown to be traceclass. The multiscattering determinant can be organized in terms of
Periodic orbits, spectral statistics, and the Riemann zeros Supersymmetry and Trace formulae: Chaos and Disorder ed J P Keating et al (New
, 1998
"... semiclassical approximation, quantum chaos, spectral statistics, the Riemann zeros I review recent developments in the semiclassical theory of spectral statistics based on the trace formula. Applications to the paircorrelation of the Riemann zeros are also discussed. ..."
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Cited by 11 (5 self)
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semiclassical approximation, quantum chaos, spectral statistics, the Riemann zeros I review recent developments in the semiclassical theory of spectral statistics based on the trace formula. Applications to the paircorrelation of the Riemann zeros are also discussed.
On the rate of quantum ergodicity on hyperbolic surfaces and billiards, Ulm report ULMTP/976, chaodyn 9707016
, 1997
"... The rate of quantum ergodicity is studied for three strongly chaotic (Anosov) systems. The quantal eigenfunctions on a compact Riemannian surface of genus g = 2 and of two triangular billiards on a surface of constant negative curvature are investigated. One of the triangular billiards belongs to th ..."
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Cited by 10 (2 self)
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The rate of quantum ergodicity is studied for three strongly chaotic (Anosov) systems. The quantal eigenfunctions on a compact Riemannian surface of genus g = 2 and of two triangular billiards on a surface of constant negative curvature are investigated. One of the triangular billiards belongs to the class of arithmetic systems. There are no peculiarities observed in the arithmetic system concerning the rate of quantum ergodicity. This contrasts to the peculiar behaviour with respect to the statistical properties of the quantal levels. It is demonstrated that the rate of quantum ergodicity in the three considered systems fits well with the known upper and lower bounds. Furthermore, Sarnak’s conjecture about quantum unique ergodicity for hyperbolic surfaces is confirmed numerically in these three systems. Note: The postscript file of this paper containing all figures is available at:
InsideOutside Duality for Planar Billiards  A Numerical Study
"... This paper reports the results of extensive numerical studies related to spectral properties of the Laplacian and the scattering matrix for planar domains (called billiards). There is a close connection between eigenvalues of the billiard Laplacian and the scattering phases, basically that every ene ..."
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Cited by 9 (1 self)
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This paper reports the results of extensive numerical studies related to spectral properties of the Laplacian and the scattering matrix for planar domains (called billiards). There is a close connection between eigenvalues of the billiard Laplacian and the scattering phases, basically that every energy at which a scattering phase is 2ß corresponds to an eigenenergy of the Laplacian. Interesting phenomena appear when the shape of the domain does not allow an extension of the eigenfunction to the exterior. In this paper these phenomena are studied and illustrated from several points of view. We consider quantum billiards, i.e., the Laplacian in a bounded domain W, with Dirichlet (zero) conditions on the boundary G. The billiard will be looked at from two different points of view, which define two seemingly independent problems. The interior problem is the more commonly studied aspect of the billiard dynamics, and the main objective in that case is to calculate the spectrum, i.e., the ei...