Results 1  10
of
10
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 47 (9 self)
 Add to MetaCart
(Show Context)
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.
Fractal Weyl laws for chaotic open systems
"... We present a result relating the density of resonances for an open chaotic system to the dimension of the classical fractal repeller of the system. The result is supported by numerical computation of the resonances of the system of n disks on a plane. The result generalizes the Weyl law for the dens ..."
Abstract

Cited by 13 (8 self)
 Add to MetaCart
(Show Context)
We present a result relating the density of resonances for an open chaotic system to the dimension of the classical fractal repeller of the system. The result is supported by numerical computation of the resonances of the system of n disks on a plane. The result generalizes the Weyl law for the density of states of a closed system to chaotic open systems.
How Chaotic is the Stadium Billiard? A Semiclassical Analysis
"... The impression gained from the literature published to date is that the spectrum of the stadium billiard can be adequately described, semiclassically, by the Gutzwiller periodic orbit trace formula together with a modified treatment of the marginally stable family of bouncing ball orbits. I show tha ..."
Abstract

Cited by 13 (0 self)
 Add to MetaCart
(Show Context)
The impression gained from the literature published to date is that the spectrum of the stadium billiard can be adequately described, semiclassically, by the Gutzwiller periodic orbit trace formula together with a modified treatment of the marginally stable family of bouncing ball orbits. I show that this belief is erroneous. The Gutzwiller trace formula is not applicable for the phase space dynamics near the bouncing ball orbits. Unstable periodic orbits close to the marginally stable family in phase space cannot be treated as isolated stationary phase points when approximating the trace of the Green function. Semiclassical contributions to the trace show an ¯h – dependent transition from hard chaos to integrable behavior for trajectories approaching the bouncing ball orbits. A whole region in phase space surrounding the marginal stable family acts, semiclassically, like a stable island with boundaries being explicitly ¯h–dependent. The localized bouncing ball states found in the billiard derive from this semiclassically stable island. The bouncing ball orbits themselves, however, do not contribute to individual eigenvalues in the spectrum. An EBK–like quantization of the regular bouncing ball eigenstates in the stadium can be derived. The stadium billiard is thus an ideal model for studying the influence of almost regular dynamics near marginally stable boundaries on quantum mechanics. This behavior is generically found at the border of classically stable islands in systems with a mixed phase space structure. PACS numbers: 05.45, 03.65.Sq 1
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 ..."
Abstract

Cited by 10 (7 self)
 Add to MetaCart
: 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...
Gutzwiller's Octagon and the Triangular Billiard T*(2,3,8) as Models for the Quantization of Chaotic Systems by Selberg's Trace Formula
, 1994
"... Two strongly chaotic systems are investigated with respect to quantization rules based on Selberg's trace formula. One of them results from the action of a particular strictly hyperbolic Fuchsian group on the Poincar'e disk, leading to a compact Riemann surface of genus g = 2. This Fuchsia ..."
Abstract

Cited by 4 (0 self)
 Add to MetaCart
Two strongly chaotic systems are investigated with respect to quantization rules based on Selberg's trace formula. One of them results from the action of a particular strictly hyperbolic Fuchsian group on the Poincar'e disk, leading to a compact Riemann surface of genus g = 2. This Fuchsian group is denoted as Gutzwiller's group. The other one is a billiard inside a hyperbolic triangle, which is generated by the operation of a reflection group denoted as T (2; 3; 8). Since both groups belong to the class of arithmetical groups, their elements can be characterized explicitly as 2 \Theta 2 matrices containing entries, which are algebraic numbers subject to a particular set of restrictions. In the case of Gutzwiller's group this property can be used to determine the geodesic length spectrum of the associated dynamical system completely up to some cutoff length. For the triangular billiard T (2; 3; 8) the geodesic length spectrum is calculated by building group elements as products ...
Quantum and Arithmetical Chaos
, 2003
"... Summary. The lectures are centered around three selected topics of quantum chaos: the Selberg trace formula, the twopoint spectral correlation functions of Riemann zeta function zeros, and of the Laplace–Beltrami operator for the modular group. The lectures cover a wide range of quantum chaos appli ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
(Show Context)
Summary. The lectures are centered around three selected topics of quantum chaos: the Selberg trace formula, the twopoint spectral correlation functions of Riemann zeta function zeros, and of the Laplace–Beltrami operator for the modular group. The lectures cover a wide range of quantum chaos applications and can serve as a nonformal introduction to mathematical methods of quantum chaos.
Classical and Quantum Transport Through Entropic Barriers Modelled by Hardwall
, 2008
"... We study the quantum transport through entropic barriers induced by hardwall constrictions of hyperboloidal shape in two and three spatial dimensions. Using the separability of the Schrödinger equation and the classical equations of motion for these geometries we study in detail the quantum transmis ..."
Abstract
 Add to MetaCart
(Show Context)
We study the quantum transport through entropic barriers induced by hardwall constrictions of hyperboloidal shape in two and three spatial dimensions. Using the separability of the Schrödinger equation and the classical equations of motion for these geometries we study in detail the quantum transmission probabilities and the associated quantum resonances, and relate them to the classical phase structures which govern the transport through the constrictions. These classical phase structures are compared to the analogous structures which, as has been shown only recently, govern reaction type dynamics in smooth systems. Although the systems studied in this paper are special due their separability they can be taken as a guide to study entropic barriers resulting from constriction geometries that lead to nonseparable dynamics.
Wavefunctions, Green’s functions and expectation values in terms of spectral determinants
, 706
"... We derive semiclassical approximations for wavefunctions, Green’s functions and expectation values for classically chaotic quantum systems. Our method consists of applying singular and regular perturbations to quantum Hamiltonians. The wavefunctions, Green’s functions and expectation values of the u ..."
Abstract
 Add to MetaCart
(Show Context)
We derive semiclassical approximations for wavefunctions, Green’s functions and expectation values for classically chaotic quantum systems. Our method consists of applying singular and regular perturbations to quantum Hamiltonians. The wavefunctions, Green’s functions and expectation values of the unperturbed Hamiltonian are expressed in terms of the spectral determinant of the perturbed Hamiltonian. Semiclassical resummation methods for spectral determinants are applied and yield approximations in terms of a finite number of classical trajectories. The final formulas have a simple form. In contrast to Poincaré surface of section methods, the resummation is done in terms of the periods of the trajectories. PACS numbers: 03.65.Sq Semiclassical theories and applications. 05.45.Mt Semiclassical chaos (“quantum chaos”).
unknown title
, 2004
"... Level dynamics in pseudointegrable billiards: an experimental study ..."
(Show Context)
Fractal Drum, Inverse . . . FOR ELLIPTIC OPERATORS AND A PARTIAL RESOLUTION OF THE WEYLBERRY CONJECTURE
 TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY
, 325
"... ..."