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Small Disks and Semiclassical Resonances

P. Rosenqvist, N. D. Whelan, A. Wirzba

Published Jan 1, 1996• 8 citations
OpenAlex

Abstract

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

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Small disks and semiclassical resonances

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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.

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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 .

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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 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 .
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 describe a wave-mechanical implementation of classically chaotic n-disk scattering based on thin two-dimensional microwave cavities. Two-, three-, and four-disk scatterings are investigated in detail. The experiments, which are able to probe the stationary Green's function of the system, yield both frequencies and widths of the low-lying quantum resonances. The observed spectra are found to be in good agreement with calculations based on semiclassical periodic orbit theory. Wave-vector autocorrelation functions are analyzed for various scattering geometries, the small wave-vector behavior allowing one to extract the escape rate from the quantum repeller. Quantitative agreement is found with the value predicted from classical scattering theory. For intermediate energies, nonuniversal oscillations are detected in the autocorrelation function, reflecting the presence of periodic orbits.
We compute the Fourier transform [ρ(L)] of the quantum-mechanical energy level density for the problem of a particle in an annular infinite well (or circular disk) of outer (inner) radius R max =RR min =fR. For various values of f=R min /R we then explicitly compare ρ(L) to the lengths of the classical periodic orbits. We also comment on the diffraction peak seen by Reimann et al. in this system, noting that at least one recurrence of that feature is present in the spectrum, and discussing the corresponding effective path lengths for these features.
The semi-classical method has become a necessary instrument to study the classical movement of the particle.Periodic orbit theory is rapidly becoming one of most useful semi-classical tools which can be used to make direct connections between the quantized energy eigenvalues of a bound state and the classical motions for the corresponding point particle.We use a quantum spectral function which contain rich information of classical orbits in well.We study the correspondence between quantum spectra and classical orbits in the circular.Two-dimensional billiard systems have provided easily visualization examples relevant for both types of analyses.As a simple example of the application to a billiard or infinite well system of Periodic orbit theory,we compute the Fourier transform(ρL) of the quantum mechanical energy level density of two-dimensional circular billiard systems.The resulting peaks in plots of ρL2 versus L are compared to the lengths of the classical trajectories in these geometries.The locations of peaks in ρL agree with the lengths of classical orbits perfectly,which testifies the correspondence of quantum mechanics and classical mechanics.This examples show evidently that semi-classical methods provides a bridge between quantum and classical mechanics.