Hyperbola Billiard Research Articles

Semiclassical theory of long time propagation in quantum chaos. Second part

We develop a semiclassical theory of wave propagation based on invariant Lagrangian manifolds existing in conservative Hamiltonian systems with chaotic dynamics. They are stable and unstable manifolds of unstable periodic orbits, and their intersections consist of homoclinic and heteroclinic orbits. For arbitrarily long times, we find matrix elements of the evolution operator between wave functions constructed in the neighbourhood of short unstable periodic orbits, in terms of canonical invariants of homoclinic and heteroclinic orbits. We verify the accuracy of these expressions by computing millions of homoclinic orbits and thousands of heteroclinic ones in the hyperbola billiard. This second part in which we describe off-diagonal matrix elements serves as complement to a previous article (first part) where we have described diagonal matrix elements.

Semiclassical theory of long time propagation in quantum chaos. First part

We develop a semiclassical theory of wave propagation based on invariant Lagrangian manifolds existing in conservative Hamiltonian systems with chaotic dynamics. They are stable and unstable manifolds of unstable periodic orbits, and their intersections consist of homoclinic and heteroclinic orbits. For arbitrary long times, we find matrix elements of the evolution operator between wave functions constructed in the neighbourhood of short unstable periodic orbits, in terms of canonical invariants of homoclinic and heteroclinic orbits. We verify the accuracy of these expressions by computing millions of homoclinic orbits and thousands of heteroclinic ones in the hyperbola billiard. Here we describe diagonal matrix elements while in another article named second part we will describe off-diagonal matrix elements.

Semiclassical quantization of highly excited scar states

The semiclassical quantization of Hamiltonian systems with classically chaotic dynamics is restricted to low excited states, close to the ground state, because the number of required periodic orbits grows exponentially with energy. Nevertheless, here we demonstrate that it is possible to find eigenenergies of highly excited states scarred by a short periodic orbit. Specifically, by using 18146 homoclinic orbits (HO)s of the shortest periodic orbit of the hyperbola billiard, we find eigenenergies of the strongest scars over a range which includes 630 even eigenfunctions. The analysis of data reveals that the used semiclassical formula presents two regimes. First, when all HOs with excursion time smaller than the Heisenberg time tH are included, the error is around 3.3% of the mean level spacing. Second, in the energy region defined by , where is the maximum excursion time included in the calculation, the error is around 15% of the mean level spacing.

Semiclassical propagator to evaluate off-diagonal matrix elements of the evolution operator between quantum states.

We present a powerful semiclassical expression to evaluate off-diagonal matrix elements of the evolution operator between quantum states constructed in the neighborhood of unstable short periodic orbits, which is valid up to the Heisenberg time. The expression is much easier to evaluate than the Van Vleck propagator and consists of a sum over the set of heteroclinic orbits, where each term of the series is computed by canonical invariants. Here we introduce relevant canonical invariants of heteroclinic orbits and with them at hand, the semiclassical expression is derived. Finally, our formula is successfully verified in the hyperbola billiard.

Semiclassical propagation up to the Heisenberg time

By using a quantum Hamiltonian system with classically chaotic dynamics, we demonstrate that it is possible to propagate waves, at a semiclassical level, for extremely long times of the order of the Heisenberg time. We achieve this unexpected result with a new formula that evaluates the autocorrelation function of a quantum state living in the neighborhood of a short periodic orbit, the so-called resonance, in terms of the set of homoclinic orbits; this set is given by the intersection of the stable and unstable manifolds of the periodic orbit. Here we study the manifolds of the shortest periodic orbit of the hyperbola billiard (a chaotic Hamiltonian system), finding a surprisingly simple tree structure. Then, we compute a complete set consisting of the first 18 146 homoclinic orbits, and by using this data we analyze the convergence of the new formula. Finally, we compare the quantum and semiclassical autocorrelation of resonances up to the Heisenberg time, obtaining a relative error O(ℏ) in correspondence with semiclassical predictions.

Ergodic properties of classical SU(2) lattice gauge theory

We investigate the relationship between the Lyapunov exponents of periodic trajectories, the average and fluctuations of Lyapunov exponents of ergodic trajectories, and the ergodic autocorrelation time for the two-dimensional hyperbola billiard. We then study the fluctuation properties of the ergodic Lyapunov spectrum of classical SU(2) gauge theory on a lattice. Our results are consistent with the notion that this system is globally hyperbolic. Among the many powerful theorems applicable to such systems, we discuss one relating to the fluctuations in the entropy growth rate.

Role of nonperiodic orbits in the semiclassical quantization of the truncated hyperbola billiard.

Based on an accurate computation of the first 1851 quantal energy levels of the truncated hyperbola billiard, we have found an anomalous long-range modulation in the integrated level density. It is shown that the observed anomaly can be explained by an additional term in Gutzwiller's trace formula. This term is given as a sum over families of closed, nonperiodic orbits which are reflected in a point of the billiard boundary where the boundary is continuously differentiable, but its curvature radius changes discontinuously.

Calculation of spectral determinants

A method for regularizing spectral determinants is developed which facilitates their computation from a finite number of eigenvalues. This is used to calcu­late the determinant ∆ for the hyperbola billiard over a range which includes 46 quantum energy levels. The result is compared with semiclassical periodic orbit evaluations of ∆ using the Dirichlet series, Euler product, and a Riemann-Siegel-type formula. It is found that the Riemann-Siegel-type expansion, which uses the least number of orbits, gives the closest approximation. This provides explicit numerical support for recent conjectures concerning the analytic proper­ties of semiclassical formulae, and in particular for the existence of resummation relations connecting long and short pseudo-orbits.

Determination of resonance spectra for bound chaotic systems

We consider the computation of the eigenvalues of the evolution operator-the resonance spectrum-by means of the zeros of a zeta function. In particular we address the problems of applying this formalism to bound chaotic systems, caused by e.g. intermittency and non-completeness of the symbolic dynamics. For bound intermittent systems we derive an approximation of the zeta function. With the aid of this zeta function it is argued that bound systems with long time tails have branch cuts in the zeta function and traces (of the evolution operator) approaching unity as a power law. We also show that the dominant time scale can be much longer than the period of the shortest periodic orbit, as is, for example, the case for the hyperbola billiard. Isolated zeros of the zeta function for the hyperbola billiard are evaluated by means of a cycle expansion. Crucial for the success of this approach is the identification of a sequence of periodic orbit responsible for a logarithmic branch cut in the zeta function. Semiclassical implications are briefly discussed.

Influence of intermittency on quantum spectra

Bound chaotic systems typically exhibit intermittency. Some semiclassical properties of intermittency are studied. The hyperbola billiard and the x2y2 model are used as strongly intermittent model systems. The almost integrable motion in the arms of the potential may be treated in the adiabatic approximation. The corresponding adiabatic Hamiltonian may be semiclassically quantized, yielding surprisingly good agreement for the hyperbola billiard. It is then demonstrated, by means of the semiclassical trace formula, how this result may be related to families of periodic orbit exclusively exploring the potential arms. It is discussed how this result implies an integrable component in the spectrum. Possible implications for the resummation problem of the trace formula, as well as for the statistical properties of energy levels, are discussed.

Pruning of orbits in four-disk and hyperbola billiards.

It is shown that for the four-disk system and the hyperbola billiard it is possible to construct a new symbolic plane preserving the orientation existing in the dynamical space. Physical orbits are mapped into the topological well-ordered plane and it is shown that the forbidden and allowed orbits are separated by a monotone pruning front.

Applications of periodic-orbit theory.

The periodic-orbit theory of Gutzwiller is applied in various ways by using generalized periodic-orbit sum rules. Numerical evaluations are carried out for the hyperbola billiard, a strongly chaotic system. The most efficient semiclassical determination of quantum energies is achieved by a quantization condition, which is formulated in terms of a zeta function by using a functional equation.

Periodic orbit quantization of bound chaotic systems

The authors present periodic orbit quantizations of the hyperbola billiard and the x2y2 potential. These two systems may be considered as belonging to the one-parameter family of potentials (x2y2)1a/. The quantum states are determined by means of the zeros of an expanded and truncated Selberg zeta function. The symmetries of the problem are considered and the Selberg zeta function is factorized into the irreducible representations of the symmetry group. The thus calculated eigenenergies are in good agreement with quantum mechanical calculations and converge when the number of terms in the expansion is increased. The results strongly indicate that the trace formula provides individual quantum eigenstates for chaotic systems.

Quantum chaos in the hyperbola billiard

A few hundred energy levels of the hyperbola billiard, a strongly chaotic system, are computed using a boundary-element method. The level statistics is investigated and found to be consistent with the predictions of random-matrix theory for the Gaussian orthogonal ensemble. The energy spectrum is used for an application of Gutzwiller's periodic orbit theory, which nicely demonstrates that the quantal energies “know” the classical periodic orbits.

Classical and quantum mechanics of a strongly chaotic billiard system

We study the hyperbola billiard, a strongly chaotic system whose classical dynamics is the free motion of a particle within the region D = l{( x, y)| x≥0∧ y≥0∧ y≤1/ x} r with elastic reflections on the boundary ∂D. The corresponding quantum mechanical problem is to determine the bound state energies as eigenvalues of the Dirichlet Laplacian on D. It is shown that the classical periodic orbits of the hyperbola billiard can be effectively enumerated by a ternary code. Combining this code with an extremum principle, we are able to determine with high precision more than 500 000 primitive periodic orbits together with their lengths, multiplicities and Lyapunov exponents. The statistical properties of the length spectrum of the periodic orbits are found to be consistent with a random walk model, which in turn predicts asymptotically an exponential proliferation of long periodic orbits and leads to a novel formula for the topological entropy τ, whose value turns out to be approximately 0.6. The periodic orbits are used for a quantitative test of Gutzwiller's periodic-orbit theory, which plays the role of a semiclassical quantization rule. We find that the predictions of the periodic-orbit theory for the Gaussian level density agree at low energies surprisingly well with the “true” results obtained from a numerical solution of the Schrödinger equation.