Use of Harmonic Inversion Techniques in the Periodic Orbit Quantization of Integrable Systems

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.Comment: 21 pages, 9 figures, submitted to Eur. Phys. J.

Higher-order hbar corrections in the semiclassical quantization of chaotic billiards

In the periodic orbit quantization of physical systems, usually only the leading-order hbar contribution to the density of states is considered. Therefore, by construction, the eigenvalues following from semiclassical trace formulae generally agree with the exact quantum ones only to lowest order of hbar. In different theoretical work the trace formulae have been extended to higher orders of hbar. The problem remains, however, how to actually calculate eigenvalues from the extended trace formulae since, even with hbar corrections included, the periodic orbit sums still do not converge in the physical domain. For lowest-order semiclassical trace formulae the convergence problem can be elegantly, and universally, circumvented by application of the technique of harmonic inversion. In this paper we show how, for general scaling chaotic systems, also higher-order hbar corrections to the Gutzwiller formula can be included in the harmonic inversion scheme, and demonstrate that corrected semiclassical eigenvalues can be calculated despite the convergence problem. The method is applied to the open three-disk scattering system, as a prototype of a chaotic system.Comment: 14 pages, 6 figures, accepted for publication in Eur. Phys. J.

Brownian Motion Model of Quantization Ambiguity and Universality in Chaotic Systems

We examine spectral equilibration of quantum chaotic spectra to universal statistics, in the context of the Brownian motion model. Two competing time scales, proportional and inversely proportional to the classical relaxation time, jointly govern the equilibration process. Multiplicity of quantum systems having the same semiclassical limit is not sufficient to obtain equilibration of any spectral modes in two-dimensional systems, while in three-dimensional systems equilibration for some spectral modes is possible if the classical relaxation rate is slow. Connections are made with upper bounds on semiclassical accuracy and with fidelity decay in the presence of a weak perturbation.Comment: 13 pages, 6 figures, submitted to Phys Rev

Semi-classical analysis of real atomic spectra beyond Gutzwiller's approximation

Real atomic systems, like the hydrogen atom in a magnetic field or the helium atom, whose classical dynamics are chaotic, generally present both discrete and continuous symmetries. In this letter, we explain how these properties must be taken into account in order to obtain the proper (i.e. symmetry projected) $\hbar$ expansion of semiclassical expressions like the Gutzwiller trace formula. In the case of the hydrogen atom in a magnetic field, we shed light on the excellent agreement between present theory and exact quantum results.Comment: 4 pages, 1 figure, final versio

Classical, semiclassical, and quantum investigations of the 4-sphere scattering system

A genuinely three-dimensional system, viz. the hyperbolic 4-sphere scattering system, is investigated with classical, semiclassical, and quantum mechanical methods at various center-to-center separations of the spheres. The efficiency and scaling properties of the computations are discussed by comparisons to the two-dimensional 3-disk system. While in systems with few degrees of freedom modern quantum calculations are, in general, numerically more efficient than semiclassical methods, this situation can be reversed with increasing dimension of the problem. For the 4-sphere system with large separations between the spheres, we demonstrate the superiority of semiclassical versus quantum calculations, i.e., semiclassical resonances can easily be obtained even in energy regions which are unattainable with the currently available quantum techniques. The 4-sphere system with touching spheres is a challenging problem for both quantum and semiclassical techniques. Here, semiclassical resonances are obtained via harmonic inversion of a cross-correlated periodic orbit signal.Comment: 12 pages, 5 figures, submitted to Phys. Rev.

Evaluation of the Productivity of a Single Subcutaneous Injection of LongRange in Stocker Calves Compared With a Positive (Dectomax) and a Negative (Saline) Control

Subclinical parasitism is commonly observed in stocker cattle. Treatment of internal parasites helps to improve weight gains, feed conversion, and immune status and decreases morbidity and mortality of beef cattle (Hawkins, 1993). Some of the most concerning classes of internal parasites include Cooperia, Haemonchus, and Ostertagia. Commonly used anthelmintics come in the form of pour-ons, oral drenches, and subcutaneous injections. A majority of these drugs are designed to be administered in a single dose and provide defense against stomach worms for approximately 14 to 42 days, but the typical grazing season lasts for approximately 120 days. For grazing cattle to have season-long protection from parasites, they may require a second dose of anthelmintic treatment, which would require cattle to be gathered and processed through a chute in the middle of the grazing season. LongRange (Merial, Duluth, GA) is the first single-dose extended release anthelmintic that provides approximately 100 to 150 days of protection. This is accomplished by combining two forms of the active ingredient: one that is released into the blood immediately after injection and a second that consists of a slow-release polymer that releases the active ingredient gradually throughout the grazing period. The objective of this study was to measure body weight productivity, fecal egg counts, and fly repellent capabilities of LongRange when administered once subcutaneously at 1.0 mg/kg body weight as a long-acting solution compared with a commercially available injectable (Dectomax; Zoetis, Florham Park, NJ) and saline in stocker cattle

The hydrogen atom in an electric field: Closed-orbit theory with bifurcating orbits

Closed-orbit theory provides a general approach to the semiclassical description of photo-absorption spectra of arbitrary atoms in external fields, the simplest of which is the hydrogen atom in an electric field. Yet, despite its apparent simplicity, a semiclassical quantization of this system by means of closed-orbit theory has not been achieved so far. It is the aim of this paper to close that gap. We first present a detailed analytic study of the closed classical orbits and their bifurcations. We then derive a simple form of the uniform semiclassical approximation for the bifurcations that is suitable for an inclusion into a closed-orbit summation. By means of a generalized version of the semiclassical quantization by harmonic inversion, we succeed in calculating high-quality semiclassical spectra for the hydrogen atom in an electric field