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49
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.
A Proof of the Gutzwiller Semiclassical Trace Formula using Coherent Sates Decomposition
 Commun. in Math. Phys
, 1999
"... The Gutzwiller trace formula links the eigenvalues of the Schrödinger operator H as Planck’s constant goes to zero (the semiclassical régime) with the closed orbits of the corresponding classical mechanical system. Gutzwiller gave a heuristic proof of this trace formula, using the Feynman integral ..."
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Cited by 35 (5 self)
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The Gutzwiller trace formula links the eigenvalues of the Schrödinger operator H as Planck’s constant goes to zero (the semiclassical régime) with the closed orbits of the corresponding classical mechanical system. Gutzwiller gave a heuristic proof of this trace formula, using the Feynman integral representation for the propagator of H. Later, using the theory of Fourier integral operators, mathematicians gave rigorous proofs of the formula in various settings. Here we show how the use of coherent states allows us to give a simple and direct proof. 1
The Semiclassical Trace Formula and Propagation of Wave Packets
 J. Funct. Anal
, 1994
"... We study spectral and propagation properties of operators of the form S ¯ h = P N j=0 ¯h j P j where 8j P j is a differential operator of order j on a manifold M , asymptotically as ¯h ! 0. The estimates are in terms of the flow fOE t g of the classical Hamiltonian H(x; p) = P N j=0 oe P j (x; ..."
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Cited by 21 (4 self)
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We study spectral and propagation properties of operators of the form S ¯ h = P N j=0 ¯h j P j where 8j P j is a differential operator of order j on a manifold M , asymptotically as ¯h ! 0. The estimates are in terms of the flow fOE t g of the classical Hamiltonian H(x; p) = P N j=0 oe P j (x; p) on T M , where oe P j is the principal symbol of P j . We present two sets of results. (I) The "semiclassical trace formula", on the asymptotic behavior of eigenvalues and eigenfunctions of S ¯ h in terms of periodic trajectories of H . (II) Associated to certain isotropic submanifolds ae T M we define families of functions f/ ¯ h g and prove that 8t fexp(\Gammait¯hS h )(/ ¯ h )g is a family of the same kind associated to OE t (). Introduction and description of results. In this paper we present some results concerning spectral and propagation properties of a class of differential operators "with small parameter", ¯h (Planck's constant). We have in mind operators of the form a(x...
Semiclassical Form Factor for Chaotic Systems With Spin 1/2
, 1999
"... . We study the properties of the twopoint spectral form factor for classically chaotic systems with spin 1/2 in the semiclassical limit, with a suitable semiclassical trace formula as our principal tool. To this end we introduce a regularized form factor and discuss the limit in which the socalled ..."
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Cited by 20 (16 self)
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. We study the properties of the twopoint spectral form factor for classically chaotic systems with spin 1/2 in the semiclassical limit, with a suitable semiclassical trace formula as our principal tool. To this end we introduce a regularized form factor and discuss the limit in which the socalled diagonal approximation can be recovered. The incorporation of the spin contribution to the trace formula requires an appropriate variant of the equidistribution principle of long periodic orbits as well as the notion of a skew product of the classical translational and spin dynamics. Provided this skew product is mixing, we show that generically the diagonal approximation of the form factor coincides with the respective predictions from random matrix theory. PACS numbers: 03.65.Sq, 05.45.Mt k Email address: bol@physik.uniulm.de  Email address: kep@physik.uniulm.de + Address after 1 October 1999: Abteilung Theoretische Physik, Universitat Ulm, AlbertEinsteinAllee 11, D89069 Ulm, G...
Between classical and quantum
, 2005
"... The relationship between classical and quantum theory is of central importance to the philosophy of physics, and any interpretation of quantum mechanics has to clarify it. Our discussion of this relationship is partly historical and conceptual, but mostly technical and mathematically rigorous, inclu ..."
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Cited by 14 (3 self)
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The relationship between classical and quantum theory is of central importance to the philosophy of physics, and any interpretation of quantum mechanics has to clarify it. Our discussion of this relationship is partly historical and conceptual, but mostly technical and mathematically rigorous, including over 500 references. For example, we sketch how certain intuitive ideas of the founders of quantum theory have fared in the light of current mathematical knowledge. One such idea that has certainly stood the test of time is Heisenberg’s ‘quantumtheoretical Umdeutung (reinterpretation) of classical observables’, which lies at the basis of quantization theory. Similarly, Bohr’s correspondence principle (in somewhat revised form) and Schrödinger’s wave packets (or coherent states) continue to be of great importance in understanding classical behaviour from quantum mechanics. On the other hand, no consensus has been reached on the Copenhagen Interpretation, but in view of the parodies of it one typically finds in the literature we describe it in detail. On the assumption that quantum mechanics is universal and complete, we discuss three ways in which classical physics has so far been believed to emerge from quantum physics, namely
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...
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.
Bifurcations of periodic orbits and uniform approximations
 J. Phys. AMath. Gen
, 1997
"... We derive uniform approximations for contributions to Gutzwiller’s periodicorbit sum for the spectral density which are valid close to bifurcations of periodic orbits in systems with mixed phase space. There, orbits lie close together and give collective contributions, while the individual contribu ..."
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Cited by 7 (2 self)
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We derive uniform approximations for contributions to Gutzwiller’s periodicorbit sum for the spectral density which are valid close to bifurcations of periodic orbits in systems with mixed phase space. There, orbits lie close together and give collective contributions, while the individual contributions of Gutzwiller’s type would diverge at the bifurcation. New results for the tangent, the period doubling and the period tripling bifurcation are given. They are obtained by going beyond the local approximation and including higher order terms in the normal form of the action. The uniform approximations obtained are tested on the kicked top and are found to be in excellent agreement with exact quantum results. PACS numbers: 03.20.+i Classical mechanics of discrete systems: general mathematical aspects. 03.65.Sq Semiclassical theories and applications. 05.45.+b Theory and models of chaotic systems. Submitted to Journal of Physics A 1 1
A Scaling Theory of Bifurcations in the Symmetric WeakNoise Escape Problem
 Journal of Statistical Physics
, 1996
"... We consider the overdamped limit of twodimensional double well systems perturbed by weak noise. In the weak noise limit the most probable fluctuational path leading from either point attractor to the separatrix (the most probable escape path, or MPEP) must terminate on the saddle between the two we ..."
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Cited by 6 (1 self)
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We consider the overdamped limit of twodimensional double well systems perturbed by weak noise. In the weak noise limit the most probable fluctuational path leading from either point attractor to the separatrix (the most probable escape path, or MPEP) must terminate on the saddle between the two wells. However, as the parameters of a symmetric double well system are varied, a unique MPEP may bifurcate into two equally likely MPEP's. At the bifurcation point in parameter space, the activation kinetics of the system become nonArrhenius. In this paper we quantify the nonArrhenius behavior of a system at the bifurcation point, by using the MaslovWKB method to construct an approximation to the quasistationary probability distribution of the system that is valid in a boundary layer near the separatrix. The approximation is a formal asymptotic solution of the Smoluchowski equation. Our analysis relies on the development of a new scaling theory, which yields `critical exponents' describing...
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 5 (3 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.