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arXiv:chao-dyn/9710012 · Journal of Physics A Mathematical and General

Periodic orbit quantization of the Sinai billiard in the small scatterer limit

Per Dahlqvist, Gábor Vattay

Published Jul 31, 1998• 8 citations

Abstract

We consider the semiclassical quantization of the Sinai billiard for disk radii R small compared with the wavelength . Via the application of the periodic orbit theory of diffraction we derive the semiclassical spectral determinant. The limitations of the derived determinant are studied by comparing it to the exact Korringa-Kohn-Rostoker (KKR) determinant, which we generalize here for the subspace. With the help of the Ewald resummation method developed for the full KKR determinant we transfer the complex diffractive determinant to a real form. The real zeros of the determinant are the quantum eigenvalues in semiclassical approximation. The essential parameter is the strength of the scatterer . Surprisingly, this can take any value between and within the range of validity of the diffractive approximation , causing strong perturbation despite the smallness of the disk. We study the statistics exhibited by spectra for fixed values of c . It is Poissonian for , provided the disk is placed inside a rectangle with irrational squared side ratio. For c = 0 we find a good agreement of the level spacing distribution with Gaussian orthogonal ensemble (GOE). The form factor and two-point correlation function are also similar to GOE, but with larger deviations. By varying the parameter c from 0 to the level statistics interpolates smoothly between these limiting cases. Any of these transitional level statistics can thus be found in the spectrum of a Sinai billiard with sufficiently small R while we go from the quantum to the semiclassical limit .

DOI: 10.1088/0305-4470/31/30/003

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We consider the semiclassical quantization of the Sinai billiard for disk radii R small compared with the wavelength . Via the application of the periodic orbit theory of diffraction we derive the semiclassical spectral determinant. The limitations of the derived determinant are studied by comparing it to the exact Korringa-Kohn-Rostoker (KKR) determinant, which we generalize here for the subspace. With the help of the Ewald resummation method developed for the full KKR determinant we transfer the complex diffractive determinant to a real form. The real zeros of the determinant are the quantum eigenvalues in semiclassical approximation. The essential parameter is the strength of the scatterer . Surprisingly, this can take any value between and within the range of validity of the diffractive approximation , causing strong perturbation despite the smallness of the disk. We study the statistics exhibited by spectra for fixed values of c . It is Poissonian for , provided the disk is placed inside a rectangle with irrational squared side ratio. For c = 0 we find a good agreement of the level spacing distribution with Gaussian orthogonal ensemble (GOE). The form factor and two-point correlation function are also similar to GOE, but with larger deviations. By varying the parameter c from 0 to the level statistics interpolates smoothly between these limiting cases. Any of these transitional level statistics can thus be found in the spectrum of a Sinai billiard with sufficiently small R while we go from the quantum to the semiclassical limit .

We study the effect on quantum spectra of the existence of small circular disks in a billiard system. In the limit where the disk radii vanish there is no effect, however this limit is approached very slowly so that even very small radii have comparatively large effects. We include diffractive orbits which scatter off the small disks in the periodic orbit expansion. This situation is formally similar to edge diffraction except that the disk radii introduce a length scale in the problem such that for wave lengths smaller than the disk radius we recover the usual semi-classical approximation; however, for wave lengths larger than the disk radius there is a qualitatively different behaviour. We test the theory by successfully estimating the positions of scattering resonances in geometries consisting of three and four small disks. PACS numbers: 0.320.+i, 03.65.Sq 1 Typeset using REVTEXThe presence of discontinuities in classical Hamiltonian systems implies the necessity of

We study the effect on quantum spectra of the existence of small circular disks in a billiard system. In the limit where the disk radii vanish there is no effect, however this limit is approached very slowly so that even very small radii have comparatively large effects. We include diffractive orbits which scatter off the small disks in the periodic orbit expansion. This situation is formally similar to edge diffraction except that the disk radii introduce a length scale in the problem such that for wavelengths smaller than the order of the disk radius we recover the usual semiclassical approximation; however, for wavelengths larger than the order of the disk radius there is a qualitatively different behaviour. We test the theory by successfully estimating the positions of scattering resonances in geometries consisting of three and four small disks.

We consider the time evolution of a wave packet representing a quantum particle moving in a geometrically open billiard that consists of a number of fixed hard-disk or hard-sphere scatterers. Using the technique of multiple collision expansions we provide a first-principle analytical calculation of the time-dependent autocorrelation function for the wave packet in the high-energy diffraction regime, in which the particle's de Broglie wavelength, while being small compared to the size of the scatterers, is large enough to prevent the formation of geometric shadow over distances of the order of the particle's free flight path. The hard-disk or hard-sphere scattering system must be sufficiently dilute in order for this high-energy diffraction regime to be achievable. Apart from the overall exponential decay, the autocorrelation function exhibits a generally complicated sequence of relatively strong peaks corresponding to partial revivals of the wave packet. Both the exponential decay (or escape) rate and the revival peak structure are predominantly determined by the underlying classical dynamics. A relation between the escape rate, and the Lyapunov exponents and Kolmogorov-Sinai entropy of the counterpart classical system, previously known for hard-disk billiards, is strengthened by generalization to three spatial dimensions. The results of the quantum mechanical calculation of the time-dependent autocorrelation function agree with predictions of the semiclassical 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 analyze the quantum spectra of the two-dimensional annular billiard system based on the open-orbit quantum spectrum function, and respectively calculate the Fourier-transformed quantum spectra for various values of a parameter （in the case f=Rin/Rout）. The results show that the peak positions of quantum spectrum match with the lengths of classical orbits of particle motion very well ，and the semiclassical closed-orbit theory affords a good explanation. While the inner circle radius becomes comparaber with the de Brogile wavelength of the particle, the character of quantum spectrum changes essentially, turning out to be similar with the pattern of optical diffraction, which is a purely wavelike phenomenon and accords with Fresnel-Kirchhoff diffraction theorem. Our treatment provides a basis theory for researching the property of billiard system and microjunction transport, and opens a new way to investigate the diffraction of crystal and spectral analysis.

For the periodic orbit quantization of a system of particles confined in a two-dimensional disk billiard, the proliferation of multiparticle periodic orbits with length precludes standard length truncation of the trace formula and alternative truncation schemes are necessary. An amplitude ordering of the single-particle periodic orbits reveals a quantum-classical correspondence for the angular momentum of a particle in the disk which is useful for the quantization of all the low energy states of the multiparticle problem. We explicitly show that a simple amplitude truncation of the trace formula better approximates the semiclassical density of states and in particular resolves closely spaced levels that are otherwise unresolved using the standard approach.