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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 ..."
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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.
Eigenvalue Spacings for Quantized Cat Maps
"... According to one of the basic conjectures in Quantum Chaos, the eigenvalues of a quantized chaotic Hamiltonian behave like the spectrum of the typical member of the appropriate ensemble of random matrices. We study one of the simplest examples of this phenomenon in the context of ergodic actions ..."
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According to one of the basic conjectures in Quantum Chaos, the eigenvalues of a quantized chaotic Hamiltonian behave like the spectrum of the typical member of the appropriate ensemble of random matrices. We study one of the simplest examples of this phenomenon in the context of ergodic actions of groups generated by several linear toral automorphisms { \cat maps". Our numerical experiments indicate that for \generic" choices of cat maps, the unfolded consecutive spacings distribution in the irreducible components of the Nth quantization (given by the Ndimensional Weil representation) approaches the GOE/GSE law of Random Matrix Theory. For certain special \arithmetic " transformations, related to the Ramanujan graphs of Lubotzky, Phillips and Sarnak, the experiments indicate that the unfolded consecutive spacings distribution follows Poisson statistics; we provide a sharp estimate in that direction.
DISSIPATION TIME AND DECAY OF CORRELATIONS
, 2003
"... Abstract. We consider the effect of noise on the dynamics generated by volumepreserving maps on a ddimensional torus. The quantity we use to measure the irreversibility of the dynamics is the dissipation time. We focus on the asymptotic behaviour of this time in the limit of small noise. We derive ..."
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Abstract. We consider the effect of noise on the dynamics generated by volumepreserving maps on a ddimensional torus. The quantity we use to measure the irreversibility of the dynamics is the dissipation time. We focus on the asymptotic behaviour of this time in the limit of small noise. We derive universal lower and upper bounds for the dissipation time in terms of various properties of the map and its associated propagators: spectral properties, local expansivity, and global mixing properties. We show that the dissipation is slow for a general class of nonweaklymixing maps; on the opposite, it is fast for a large class of exponentially mixing systems which include uniformly expanding maps and Anosov diffeomorphisms. 1.
DISSIPATION TIME OF QUANTIZED TORAL MAPS.
, 2005
"... Abstract. We introduce the notion of dissipation time for noisy quantum maps on the 2ddimensional torus as an analogue of its previously studied classical version. We show that this dissipation time is sensitive to the chaotic behavior of the corresponding classical system, if one simultaneously con ..."
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Abstract. We introduce the notion of dissipation time for noisy quantum maps on the 2ddimensional torus as an analogue of its previously studied classical version. We show that this dissipation time is sensitive to the chaotic behavior of the corresponding classical system, if one simultaneously considers the semiclassical limit ( � → 0) together with the limit of small noise strength (ǫ → 0). Focusing on quantized smooth Anosov maps, we exhibit a semiclassical régime � < ǫ E ≪ 1 (where E> 1) in which classical and quantum dissipation times share the same asymptotics: in this régime, a quantized Anosov map dissipates fast, as the classical map does. As an intermediate result, we obtain rigorous estimates of the quantumclassical correspondence for noisy maps on the torus, up to times logarithmic in � −1. On the other hand, we show that in the “quantum régime ” ǫ ≪ � ≪ 1, quantum and classical dissipation times behave very differently. In the special case of ergodic toral symplectomorphisms (generalized “Arnold’s cat ” maps), we obtain the exact asymptotics of the quantum dissipation time and precise the régime of correspondence between quantum and classical dissipations. 1.
Online at stacks.iop.org/Non/17/1481
, 2004
"... View the table of contents for this issue, or go to the journal homepage for more ..."
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Relaxation Time of Quantized Toral Maps
"... Abstract. We introduce the notion of relaxation time for noisy quantum maps on the 2ddimensional torus – generalization of previously studied dissipation time. We show that the relaxation time is sensitive to the chaotic behavior of the corresponding classical system if one simultaneously considers ..."
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Abstract. We introduce the notion of relaxation time for noisy quantum maps on the 2ddimensional torus – generalization of previously studied dissipation time. We show that the relaxation time is sensitive to the chaotic behavior of the corresponding classical system if one simultaneously considers the semiclassical limit ( � → 0) together with the limit of small noise strength (ɛ → 0). Focusing on quantized smooth Anosov maps, we exhibit a semiclassical régime � <ɛ E ≪ 1(whereE>1) in which classical and quantum relaxation times share the same asymptotics: in this régime, a quantized Anosov map relaxes to equilibrium fast, as the classical map does. As an intermediate result, we obtain rigorous estimates of the quantumclassical correspondence for noisy maps on the torus, up to times logarithmic in � −1. On the other hand, we show that in the “quantum régime ” ɛ ≪ � ≪ 1, quantum and classical relaxation times behave very differently. In the special case of ergodic toral symplectomorphisms (generalized “Arnold’s cat” maps), we obtain the exact asymptotics of the quantum relaxation time and precise the régime of correspondence between quantum and classical relaxations. 1