Periodic orbit theory of diffraction.

The semiclassical limit of the quantum few-body problem has not been studied in general terms from the point of view of periodic orbit theory. In a previous paper, we studied noninteracting two-body systems [Phys. Rev. A 62, 042109 (2000)] and discussed the fact that the periodic orbits occur in continuous families. Interactions destroy the periodic orbit families leaving a discrete set of isolated periodic orbits. In this paper, we consider the effect of weak two-body interactions, which can be thought of as symmetry-breaking perturbations and can thus be analyzed using a theory developed by Creagh [Ann. Phys. (N.Y.) 248, 1 (1996)]. The P\"oschl-Teller two-body system confined in a square well is analyzed to illustrate the use of the formalism. It is shown that the effect of the interaction can be evaluated for all two-particle periodic orbits, and that the coarse-grained quantum density of states can be fully reproduced from simply summing the perturbed contributions of each periodic orbit family. Good numerical estimates of the quantum singlet energies can actually be obtained, but it is found that that perturbed trace formulas cannot reproduce the multiplet splittings predicted from quantum mechanics. Several interesting properties are observed depending on the range of the interaction and on whether the interaction is attractive or repulsive.

We investigate the resonance spectrum of the Hénon-Heiles potential up to twice the barrier energy. The quantum spectrum is obtained by the method of complex coordinate rotation. We use periodic orbit theory to approximate the oscillating part of the resonance spectrum semiclassically and Strutinsky smoothing to obtain its smooth part. Although the system in this energy range is almost chaotic, it still contains stable periodic orbits. Using Gutzwiller's trace formula, complemented by a uniform approximation for a co-dimension-two bifurcation scenario, we are able to reproduce the coarse-grained quantum-mechanical density of states very accurately, including only a few stable and unstable orbits.

Path integration over closed loops and Gutzwiller's trace formula

We investigate the resonance spectrum of the H\\\'enon-Heiles potential up to twice the barrier energy. The quantum spectrum is obtained by the method of complex coordinate rotation. We use periodic orbit theory to approximate the oscillating part of the resonance spectrum semiclassically and Strutinsky smoothing to obtain its smooth part. Although the system in this energy range is almost chaotic, it still contains stable periodic orbits. Using Gutzwiller\'s trace formula, complemented by a uniform approximation for a codimension-two bifurcation scenario, we are able to reproduce the coarse-grained quantum-mechanical density of states very accurately, including only a few stable and unstable orbits.

Harmonic inversion has already been proven to be a powerful tool for the analysis of quantum spectra and the periodic orbit orbit quantization of chaotic systems. The harmonic inversion technique circumvents the convergence problems of the periodic orbit sum and the uncertainty principle of the usual Fourier analysis, thus yielding results of high resolution and high precision. Based on the close analogy between periodic orbit trace formulae for regular and chaotic systems the technique is generalized in this paper for the semiclassical quantization of integrable systems. Thus, harmonic inversion is shown to be a universal tool which can be applied to a wide range of physical systems. The method is further generalized in two directions: Firstly, the periodic orbit quantization will be extended to include higher order hbar corrections to the periodic orbit sum. Secondly, the use of cross-correlated periodic orbit sums allows us to significantly reduce the required number of orbits for semiclassical quantization, i.e., to improve the efficiency of the semiclassical method. As a representative of regular systems, we choose the circle billiard, whose periodic orbits and quantum eigenvalues can easily be obtained.

Classical and quantum mechanics of a strongly chaotic billiard system

A nonrelativistic superparticle moving freely on the super-Riemann surface of genus g≥2 is investigated. The classical motion is characterized by the super-Fuchsian group. The periodic orbits are unstable and the system is classically chaotic. Quantization is performed in the path integral method. The kernel function is explicitly given in the semiclassical approximation. The quantized energy spectrum is related to the length spectrum of the classical periodic orbits, which is a superanalog of Selberg’s trace formula.

For a modified Kepler problem, we reexamine jumps in the saturation spectral rigidity and large oscillations of the level number variance with near zero minima. Earlier discrepancy between the periodic orbit theory and numerical calculation is cleared by a quantum mechanical calculation. A new class of radial periodic orbits is included establishing a complete correspondence between the periodic orbit theory and the quantum mechanical approach. We show that the diagonal approximation for the level density in the periodic orbit theory already gives a good fit with the numerical calculation. Even greater accuracy is achieved by considering coherent interference between the classical periodic orbits term and the Balian-Bloch term. This procedure produces improved results for the hard-wall rectangular billiards as well.

Energy level statistics of quantized chaotic systems have been evaluated in the semiclassical limit via their periodic orbits using the Gutzwiller and related trace formulae. Here we evaluate a spectral statistic of chaotic 4-regular quantum graphs from their periodic orbits without the semiclassical limit. The variance of the n-th coefficient of the characteristic polynomial is determined by the sizes of the sets of distinct primitive periodic orbits with n bonds which have no self-intersections, and the sizes of the sets with a given number of self-intersections which all consist of two sections of the pseudo orbit crossing at a single vertex. Using this result we observe the mechanism that connects semiclassical results to the total number of orbits regardless of their structure.

We study the classical and quantum correspondence for a two-dimensional Sinai billiard system. By using the Stationary state expansion method and Gutzwiller's periodic orbit theory, we analyze the quantum length spectrum obtained through the Fourier transformation of the quantum density of state for the Sinai billiard system, and by comparing the peak position with the length of the classical periodic orbit we find their excellent correspondence. We observe that some quantum states are localized near some short period orbits, forming the quantum scarred states or superscarred states. In this paper we also investigate the nearest-neighbor spacing distribution of levels for both concentric and nonconcentric Sinai billiard systems, and find that the concentric Sinai billiard system is nearintegrable, and for the nonconcentric Sinai billiard system with =3/8 its nearest-neighbor spacing distribution of levels transits from nearintegrable to the Wigner distribution as the distance between the two centers increases.

Vertical thermal convection exhibits weak turbulence and spatio-temporally chaotic behaviour. For this configuration, we report seven new equilibria and 26 new periodic orbits. These orbits, together with four previously studied in Zheng et al. ( J. Fluid Mech. , 2024 b , vol. 1000, p. A29) bring the number of periodic-orbit branches computed so far to 30, all solutions to the fully nonlinear three-dimensional Navier–Stokes equations. These new and unstable invariant solutions capture intricate spatio-temporal flow patterns including straight, oblique, wavy, skewed and distorted convection rolls, as well as bursts and defects. These interesting and important fluid mechanical processes in a small flow unit are shown to also appear locally and instantaneously in a chaotic simulation in a large domain. Most of the solution branches show rich spatial and/or spatio-temporal symmetries. The bifurcation-theoretic organisation of these solutions is discussed; the bifurcation scenarios include Hopf, pitchfork, saddle-node, period-doubling, period-halving, global homoclinic and heteroclinic bifurcations, as well as isolas. Furthermore, these orbits are shown to be able to reconstruct statistically the core part of the attractor, so that these results may contribute to a quantitative description of transitional fluid turbulence using periodic orbit theory.

Periodic orbit (PO) theory can be used to make connections between the quantum energy eigenvalue spectrum and the closed orbits of the corresponding classical system. The two-dimensional annular billiard or circular disk system (namely, a particle in the plane confined between inner and outer infinite circular walls at fR≡Rin<Rout≡R where 0<f<1) is examined in the context of periodic orbit theory to illustrate several novel aspects of the PO analysis. Important features of this problem include (i) the appearance and disappearance of various features in the classical path length spectrum as a parameter (in this case f=Rin/Rout), is continuously varied, (ii) the presence of path length features which do not correspond to classical trajectories, but are rather due to purely wavelike phenomena (namely, diffraction around the inner annulus), and (iii) the study of the contribution of different regions in the energy eigenvalue space to different classes of classical trajectories. This last feature is a general observation about periodic orbit theory which can be usefully applied to a number of billiard systems. In addition, since the annular disk system is seldom studied in detail in either classical or quantum mechanics, we provide an appendix which reviews some of the more traditional connections between the classical and quantum probability densities for this geometry, exemplifying some of the important differences between one- and two-dimensional systems which arise from purely geometrical rather than kinematical effects.

We discuss a basic mathematical approach to Feynman path integrals as infinite-dimensional oscillatory integrals. We present new results on asymptotics of such integrals which exploit recently developed approximation techniques via finite dimensional oscillatory integrals. Applications are also given, namely to the study of the trace of the time evolution operator in quantum mechanics and to the interpretation of “Gutzwiller's trace formula” as a leading term in an asymptotic expansion around classical periodic orbits.

We derive semiclassical trace formulae including Gutzwiller's trace formula using coherent states. This formulation has several advantages over the usual coordinate‐space formulation. Using a coherent‐state basis makes it immediately obvious that classical periodic orbits make separate contributions to the trace of the quantum‐mechanical time evolution operator. In addition, our approach is manifestly canonically invariant at all stages, and leads to the simplest possible derivation of Gutzwiller's formula.

For the reaction of He with H2+, starting with accurate theoretically computed reactive, elastic, and inelastic scattering data that reveals many complex unassignable narrow resonances, the detailed motions governing the dynamics of the tight transition state are extracted. Methods ranging from scattering theory, the stabilization theory of dynamics, nonlinear dynamic periodic orbit theory, and hierarchical smoothing theory which was developed to study complex ‘‘chaotic’’ spectra, are all used in the analysis. Relationships between the doorway model of nuclear physics, aspects of transition state theory, and models of nonlinear chaotic dynamics are pointed out and used to shed light on the fact that the complex resonance structure observed is one quantum manifestation of classical transient chaos in scattering processes. The transition (or doorway) state corresponds to the only populous and robust periodic orbit or set of similar periodic orbits whose motion allows the types of energy transfers necessary to go from reactants to products. Wave packet motion and quantum eigenfunctions are influenced by these periodic orbits.