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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 ..."
Abstract

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.
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
The Missing Link Between the Quantum Mechanical and SemiClassical Determination of Scattering Resonance Poles
, 1996
"... We investigate the twodimensional scattering of a pointparticle from n nonoverlapping fixed disks and study  using these systems  the connection between the spectral properties of the quantum mechanical scattering matrix and its semiclassical equivalent based on the GutzwillerVoros zetafunc ..."
Abstract
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We investigate the twodimensional scattering of a pointparticle from n nonoverlapping fixed disks and study  using these systems  the connection between the spectral properties of the quantum mechanical scattering matrix and its semiclassical equivalent based on the GutzwillerVoros zetafunction. We rewrite the determinant of the scattering matrix 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 n disk system  and the ratio of two mutually complex conjugate determinants of the genuinely multiscattering kernel, M, of KKRtype which represents the coherent part of the ndisk scattering. Our result is welldefined at every step of the calculation, as the onshell Tmatrix and the kernel A = M\Gamma1 are shown to be traceclass. We stress that the cumulant expansion (which defines the determinant over an infinite, but trace class matrix) imposes the curvature regula...