From classical periodic orbits to the quantization of chaos

1992 ◽  
Vol 437 (1901) ◽  
pp. 693-714 ◽  
Keyword(s):  
Periodic Orbit ◽  
Zeta Function ◽  
Dirichlet Series ◽  
Periodic Orbits ◽  
Trace Formula ◽  
Riemann Surfaces ◽  
Chaotic Systems ◽  
Critical Line ◽  
Selberg Zeta Function ◽  
Periodic Orbit Theory

We present a quantitative analysis of Selberg’s trace formula viewed as an exact version of Gutzwiller’s semiclassical periodic-orbit theory for the quantization of classically chaotic systems. Two main applications of the trace formula are discussed in detail, (i) The periodic-orbit sum rules giving a smoothing of the quantal energy-level density. (ii) The Selberg zeta function as a prototype of a dynamical zeta function defined as an Euler product over the classical periodic orbits and its analytic continuation across the entropy barrier by means of a Dirichlet series. It is shown how the long periodic orbits can be effectively taken into account by a universal remainder term which is explicitly given as an integral over an ‘orbit-selection function’. Numerical results are presented for the free motion of a point particle on compact Riemann surfaces (Hadamard-Gutzwiller model), which is the primary testing ground for our ideas relating quantum mechanics and classical mechanics in the case of strong chaos. Our results demonstrate clearly the crucial role played by the long periodic orbits. An exact rule for quantizing chaos is derived for such systems where the Dirichlet series representing the Selberg zeta function converges on the critical line. Explicit formulae are given for the computation of the abscissae of absolute and conditional convergence, respectively, of these dynamical Dirichlet series. For the two Riemann surfaces considered, it turns out that one can cross the entropy barrier, but that the critical line cannot be reached by a convergent Dirichlet series. It would seem that this is the main reason why the Riemann-Siegel lookalike formula, recently conjectured by M. V. Berry and J. P. Keating, fails in generating the lower-lying quantal energies for these strongly chaotic systems.

A new asymptotic representation for ζ(½ + i t ) and quantum spectral determinants

1992 ◽  
Vol 437 (1899) ◽  
pp. 151-173 ◽  
Keyword(s):  
Zeta Function ◽  
Analytic Continuation ◽  
Dirichlet Series ◽  
Asymptotic Representation ◽  
Critical Line ◽  
Error Function ◽  
Real Variable ◽  
Computational Effort ◽  
Selberg Zeta Function ◽  
Semiclassical Approximations

By analytic continuation of the Dirichlet series for the Riemann zeta function ζ(s) to the critical line s = ½ + i t ( t real), a family of exact representations, parametrized by a real variable K , is found for the real function Z ( t ) = ζ(½ + i t ) exp {iθ( t )}, where θ is real. The dominant contribution Z 0 ( t,K ) is a convergent sum over the integers n of the Dirichlet series, resembling the finite ‘main sum ’ of the Riemann-Siegel formula (RS) but with the sharp cut-off smoothed by an error function. The corrections Z 3 ( t,K ), Z 4 ( t,K )... are also convergent sums, whose principal terms involve integers close to the RS cut-off. For large K , Z 0 contains not only the main sum of RS but also its first correction. An estimate of high orders m ≫ 1 when K < t 1/6 shows that the corrections Z k have the ‘factorial/power ’ form familiar in divergent asymptotic expansions, the least term being of order exp { ─½ K 2 t }. Graphical and numerical exploration of the new representation shows that Z 0 is always better than the main sum of RS, providing an approximation that in our numerical illustrations is up to seven orders of magnitude more accurate with little more computational effort. The corrections Z 3 and Z 4 give further improvements, roughly comparable to adding RS corrections (but starting from the more accurate Z 0 ). The accuracy increases with K , as do the numbers of terms in the sums for each of the Z m . By regarding Planck’s constant h as a complex variable, the method for Z ( t ) can be applied directly to semiclassical approximations for spectral determinants ∆( E, h ) whose zeros E = E j ( h ) are the energies of stationary states in quantum mechanics. The result is an exact analytic continuation of the exponential of the semiclassical sum over periodic orbits given by the divergent Gutzwiller trace formula. A consequence is that our result yields an exact asymptotic representation of the Selberg zeta function on its critical line.

The Riemann-Siegel expansion for the zeta function: high orders and remainders

1995 ◽  
Vol 450 (1939) ◽  
pp. 439-462 ◽  
Keyword(s):  
Zeta Function ◽  
Dirichlet Series ◽  
Critical Line ◽  
High Accuracy ◽  
Stokes Line ◽  
Power Dependence ◽  
Multiplier Function ◽  
Riemann’S Zeta Function

On the critical line s ═ ½ + i t ( t real), Riemann’s zeta function can be calculated with high accuracy by the Riemann-Siegel expansion. This is derived here by elementary formal manipulations of the Dirichlet series. It is shown that the expansion is divergent, with the high orders r having the familiar 'factorial' divided by power' dependence, decorated with an unfamiliar slowly varying multiplier function which is calculated explicitly. Terms of the series decrease until r ═ r * ≈ 2π t and then increase. The form of the remainder when the expansion is truncated near r * is determined; it is of order exp(-π t ), indicating that the critical line is a Stokes line for the Riemann-Siegel expansion. These conclusions are supported by computations of the first 50 coefficients in the expansion, and of the remainders as a function of truncation for several values of t .

Quantum chaos and quantum graphs

2018 ◽  
Author(s):  
Jac Verbaarschot
Keyword(s):  
Periodic Orbit ◽  
Quantum Chaos ◽  
Sigma Model ◽  
Chaotic Systems ◽  
Semiclassical Approximation ◽  
Quantum Evolution ◽  
Quantum Graphs ◽  
Periodic Orbit Theory ◽  
Spectral Statistics ◽  
Orbit Theory

This article examines the origins of the universality of the spectral statistics of quantum chaotic systems in the context of periodic orbit theory. It also considers interesting analogies between periodic orbit theory and the sigma model, along with related work on quantum graphs. The article first reviews some facts and definitions for classically chaotic systems in order to elucidate their quantum behaviour, focusing on systems with two degrees of freedom: one characterized by ergodicity and another by hyperbolicity. It then describes two semiclassical approximation techniques — Gutzwiller’s periodic orbit theory and a refined approach incorporating the unitarity of the quantum evolution — and highlights their importance in understanding universal spectral statistics, and how they are related to the sigma model. This is followed by an analysis of parallel developments for quantum graphs, which are relevant to quantum chaos.

QUANTIZATION RULES FOR STRONGLY CHAOTIC SYSTEMS

1992 ◽  
Vol 06 (27) ◽  
pp. 1691-1719 ◽  
Author(s):  
R. AURICH ◽  
J. BOLTE
Keyword(s):  
Periodic Orbit ◽  
Chaotic Systems ◽  
Level Density ◽  
Quantization Condition ◽  
Periodic Orbit Theory ◽  
Symmetry Classes ◽  
Dynamical Zeta Function ◽  
The Mean ◽  
Orbit Theory ◽  
Quantization Rules

We discuss the quantization of strongly chaotic systems and apply several quantization rules to a model system given by the unconstrained motion of a particle on a compact surface of constant negative Gaussian curvature. We study the periodic-orbit theory for distinct symmetry classes corresponding to a parity operation which is always present when such a surface has genus two. Recently, several quantization rules based on periodic orbit theory have been introduced. We compare quantizations using the dynamical zeta function Z(s) with the quantization condition [Formula: see text] where a periodic-orbit expression for the spectral staircase N(E) is used. A general discussion of the efficiency of periodic-orbit quantization then allows us to compare the different methods. The system dependence of the efficiency, which is determined by the topological entropy τ and the mean level density [Formula: see text], is emphasized.

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