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18
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.
Semiclassical Transition from an Elliptical to an Oval
 Billiard, J. Phys. A
, 1997
"... Semiclassical approximations often involve the use of stationary phase approximations. This method can be applied when ¯h is small in comparison to relevant actions or action differences in the corresponding classical system. In many situations, however, action differences can be arbitrarily small a ..."
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Cited by 7 (3 self)
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Semiclassical approximations often involve the use of stationary phase approximations. This method can be applied when ¯h is small in comparison to relevant actions or action differences in the corresponding classical system. In many situations, however, action differences can be arbitrarily small and then uniform approximations are more appropriate. In the present paper we examine different uniform approximations for describing the spectra of integrable systems and systems with mixed phase space. This is done on the example of two billiard systems, an elliptical billiard and a deformation of it, an oval billiard. We derive a trace formula for the ellipse which involves a uniform approximation for the Maslov phases near the separatrix, and a uniform approximation for tori of periodic orbits close to a bifurcation. We then examine how the trace formula is modified when the ellipse is deformed into an oval. This involves uniform approximations for the breakup of tori and uniform approximations for bifurcations of periodic orbits. Relations between different uniform approximations are discussed. PACS numbers: 03.65.Ge Solutions of wave equations: bound states. 03.65.Sq Semiclassical theories and applications. 05.45.+b Theory and models of chaotic systems.
Triaxial ellipsoidal quantum billiards. Ann. Phys
, 1999
"... The classical mechanics, exact quantum mechanics and semiclassical quantum mechanics of the billiard in the triaxial ellipsoid is investigated. The system is separable in ellipsoidal coordinates. A smooth description of the motion is given in terms of a geodesic flow on a solid torus, which is a fou ..."
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Cited by 7 (4 self)
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The classical mechanics, exact quantum mechanics and semiclassical quantum mechanics of the billiard in the triaxial ellipsoid is investigated. The system is separable in ellipsoidal coordinates. A smooth description of the motion is given in terms of a geodesic flow on a solid torus, which is a fourfold cover of the interior of the ellipsoid. Two crossing separatrices lead to four generic types of motion. The action variables of the system are integrals of a single Abelian differential of second kind on a hyperelliptic curve of genus 2. The classical separability carries over to quantum mechanics giving two versions of generalized Lamé equations according to the two sets of classical coordinates. The quantum eigenvalues define a lattice when transformed to classical action space. Away from the separatrix surfaces the lattice is given by EBK quantization rules for the four types of classical motion. The transition between the four lattices is described by a uniform semiclassical quantization scheme based on a WKB ansatz. The tunneling between tori is given by penetration integrals which again are integrals of the same Abelian differential that gives the classical action variables. It turns out that the quantum mechanics of ellipsoidal billiards is semiclassically most elegantly explained by the investigation of its hyperelliptic curve and the real and purely imaginary periods of a single Abelian differential.
Energy Surfaces of Ellipsoidal Billiards
 Z. Naturforsch
, 1996
"... Energy surfaces in the space of action variables are calculated and graphically presented for general triaxial ellipsoidal billiards. As was demonstrated by Jacobi in 1838, the system may be integrated in terms of hyperelliptic functions. The actual computation, however, has never been done. It is f ..."
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Cited by 5 (5 self)
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Energy surfaces in the space of action variables are calculated and graphically presented for general triaxial ellipsoidal billiards. As was demonstrated by Jacobi in 1838, the system may be integrated in terms of hyperelliptic functions. The actual computation, however, has never been done. It is found that generic energy surfaces consist of seven pieces, representing topologically different types of invariant tori. The character of the corresponding motion is discussed. Frequencies, winding numbers, and the location of resonances are also determined. The results may serve as a basis for perturbation theory of slightly modified systems, and for semiclassical quantization. 1 1 Introduction Hamiltonian systems with three degrees of freedom have recently attracted considerable attention. Even though the interest is primarily directed towards nonintegrable dynamics, it is important to have integrable limiting cases available as a reference. Ellipsoidal billiards are highly nontrivial...
The elliptic quantum billiard
 Ann. Phys
, 1997
"... The exact and semiclassical quantum mechanics of the elliptic billiard is investigated. The classical system is integrable and exhibits a separatrix, dividing the phase space into regions of oscillatory and rotational motion. The classical separability carries over to quantum mechanics, and the Schr ..."
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Cited by 5 (3 self)
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The exact and semiclassical quantum mechanics of the elliptic billiard is investigated. The classical system is integrable and exhibits a separatrix, dividing the phase space into regions of oscillatory and rotational motion. The classical separability carries over to quantum mechanics, and the Schrödinger equation is shown to be equivalent to the spheroidal wave equation. The quantum eigenvalues show a clear pattern when transformed into the classical action space. The implication of the separatrix on the wave functions is illustrated. A uniform WKB quantization taking into account complex orbits is shown to be adequate for the semiclassical quantization in the presence of a separatrix. The pattern of states in classical action space is nicely explained by this quantization procedure. We extract an effective Maslov phase varying smoothly on the energy surface, which is used to modify the BerryTabor trace formula, resulting in a summation over nonperiodic orbits. This modified trace formula produces the correct number of states, even close to the separatrix. The Fourier transform of the density of states is explained in terms of classical orbits, and the The semiclassical quantization of a Hamiltonian system is deeply connected to the structure of its phase space. The generic Hamiltonian system contains a complicated mixture of nearintegrable and
A TwoParameter Study of the Extent of Chaos in a Billiard System
 Chaos
, 1995
"... The billiard system of Benettin and Strelcyn [2] is generalized to a twoparameter family of different shapes. Its boundaries are composed of circular segments. The family includes the integrable limit of a circular boundary, convex boundaries of various shapes with mixed dynamics, stadiums, and a v ..."
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Cited by 2 (2 self)
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The billiard system of Benettin and Strelcyn [2] is generalized to a twoparameter family of different shapes. Its boundaries are composed of circular segments. The family includes the integrable limit of a circular boundary, convex boundaries of various shapes with mixed dynamics, stadiums, and a variety of nonconvex boundaries, partially with ergodic behaviour. The extent of chaos has been measured in two ways: (i) in terms of phase space volume occupied by the main chaotic band, and (ii) in terms of the Lyapunov exponent of that same region. The results are represented as a kind of phase diagram of chaos. We observe complex regularities, related to the bifurcation scheme of the most prominent resonances. A detailed stability analysis of these resonances up to period six explains most of these features. The phenomenon of breathing chaos [13], observed earlier in a oneparameter study of the gravitational wedge billiard, is part of the picture, giving support to the conjecture that ...
Coincidence of length spectra does not imply isospectrality
, 2008
"... Penrose–Lifshits mushrooms are planar domains coming in nonisometric pairs with the same geodesic length spectrum. Recently S. Zelditch raised the question whether such billiards also have the same eigenvalue spectrum for the Dirichlet Laplacian (conjecturing “no”). Here we show that generically (in ..."
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Cited by 1 (0 self)
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Penrose–Lifshits mushrooms are planar domains coming in nonisometric pairs with the same geodesic length spectrum. Recently S. Zelditch raised the question whether such billiards also have the same eigenvalue spectrum for the Dirichlet Laplacian (conjecturing “no”). Here we show that generically (in the class of smooth domains) the two members of a mushroom pair have different spectra. 1
The First Birkhoff Coefficient and the Stability of 2Periodic Orbits on Billiards
, 2004
"... In this work we address the question of proving the stability of elliptic 2periodic orbits for strictly convex billiards. Eventhough it is part of a widely accepted belief that ellipticity implies stability, classical theorems show that the certainty of stability relies upon more fine conditions. W ..."
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In this work we address the question of proving the stability of elliptic 2periodic orbits for strictly convex billiards. Eventhough it is part of a widely accepted belief that ellipticity implies stability, classical theorems show that the certainty of stability relies upon more fine conditions. We present a review of the main results and general theorems and describe the procedure to fullfill the supplementary conditions for strictly convex billiards. 1
ATOMOPTICS BILLIARDS Nonlinear dynamics with cold atoms in optical traps
, 2004
"... We present a new experimental system (the “atomoptics billiard”) and demonstrate chaotic and regular dynamics of cold, optically trapped atoms. We show that the softness of the walls and additional optical potentials can be used to manipulate the structure of phase space. ..."
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We present a new experimental system (the “atomoptics billiard”) and demonstrate chaotic and regular dynamics of cold, optically trapped atoms. We show that the softness of the walls and additional optical potentials can be used to manipulate the structure of phase space.
Fac. de Ingeniería, Uruguay. Sylvie Oliffson Kamphorst † Sônia Pinto de Carvalho
, 1995
"... The elliptical stadium is a curve constructed by joining two halfellipses, with half axes a> 1 and b = 1, by two straight segments of equal length 2h. Donnay [6] has shown that if 1 < a < √ 2 and if h is big enough, then the corresponding billiard map has a positive Lyapunov exponent almos ..."
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The elliptical stadium is a curve constructed by joining two halfellipses, with half axes a> 1 and b = 1, by two straight segments of equal length 2h. Donnay [6] has shown that if 1 < a < √ 2 and if h is big enough, then the corresponding billiard map has a positive Lyapunov exponent almost everywhere; moreover, h → ∞ as a → √ 2. 4 − 2 2, then h> 2a 2 √ a 2 − 1 assures the positiveness In this work we prove that if 1 < a < of a Lyapunov exponent. And we conclude that, for these values of a and h, the elliptical stadium billiard mapping is ergodic and has the Kproperty. 1