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43
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 40 (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 34 (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
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 quant ..."
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Cited by 13 (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
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 12 (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
Inverse spectral problem for analytic domains III: domains with corners (preprint
, 2001
"... Abstract. This paper develops and implements a new algorithm for calculating wave trace invariants of a bounded plane domain around a periodic billiard orbit. The algorithm is based on a new expression for the localized wave trace as a special multiple oscillatory integral over the boundary, and on ..."
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Cited by 11 (2 self)
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Abstract. This paper develops and implements a new algorithm for calculating wave trace invariants of a bounded plane domain around a periodic billiard orbit. The algorithm is based on a new expression for the localized wave trace as a special multiple oscillatory integral over the boundary, and on a Feynman diagrammatic analysis of the stationary phase expansion of the oscillatory integral. The algorithm is particularly effective for Euclidean plane domains possessing a Z2 symmetry which reverses the orientation of a bouncing ball orbit. It is also very effective for domains with dihedral symmetries. For simply connected analytic Euclidean plane domains in either symmetry class, we prove that the domain is determined within the class by either its Dirichlet or Neumann spectrum. This improves and generalizes the previous best inverse result of [Z1, Z2, ISZ] that simply connected analytic plane domains with two symmetries are spectrally determined within that class. 1.
Inverse spectral problem for analytic plane domains IV (in preparation
 Department of Mathematics, Johns Hopkins University
"... Abstract. This is the first in a series of papers [Z3, Z4, Z5, Z6, Z7] on inverse spectral/resonance problems for analytic plane domains Ω. In this paper, we present a rigorous version of the BalianBloch trace formula [BB1, BB2]. It is an asymptotic formula for the trace TrRρ(k + iτ) of the regular ..."
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Cited by 7 (3 self)
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Abstract. This is the first in a series of papers [Z3, Z4, Z5, Z6, Z7] on inverse spectral/resonance problems for analytic plane domains Ω. In this paper, we present a rigorous version of the BalianBloch trace formula [BB1, BB2]. It is an asymptotic formula for the trace TrRρ(k + iτ) of the regularized resolvent of the Dirichlet Laplacian of Ω as k → ∞ with τ> 0 held fixed. When the support of ˆρ contains the length Lγ of precisely one periodic reflecting ray γ, then the asymptotic expansion of TrRρ(k + iτ) is essentially the same as the wave trace expansion at γ. The raison d’etre for this approach to wave invariants is that they are explicitly computable. Applications of the trace formula will be given in the subsequent articles in this series. For instance, in [Z4, Z5] we will prove that analytic domains with one symmetry are determined by their Dirichlet spectra. Although we only present details in dimension 2, the BalianBloch approach works the same in all dimensions. 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.
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
Estimates from below for the spectral function and for the remainder in local Weyl's law
"... Abstract. We obtain asymptotic lower bounds for the spectral function of the Laplacian and for the remainder in local Weyl’s law on manifolds. In the negatively curved case, thermodynamic formalism is applied to improve the estimates. Key ingredients of the proof include the wave equation parametrix ..."
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Cited by 6 (4 self)
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Abstract. We obtain asymptotic lower bounds for the spectral function of the Laplacian and for the remainder in local Weyl’s law on manifolds. In the negatively curved case, thermodynamic formalism is applied to improve the estimates. Key ingredients of the proof include the wave equation parametrix, a pretrace formula and the Dirichlet box principle. Our results develop and extend the unpublished thesis of A. Karnaukh [K]. 1. Introduction and