624 research outputs found
Inclusion of Diffraction Effects in the Gutzwiller Trace Formula
The Gutzwiller trace formula is extended to include diffraction effects. The
new trace formula involves periodic rays which have non-geometrical segments as
a result of diffraction on the surfaces and edges of the scatter.Comment: 4 pages, LaTeX, 1 ps figur
Wave Chaos in Elastodynamic Cavity Scattering
The exact elastodynamic scattering theory is constructed to describe the
spectral properties of two- and more-cylindrical cavity systems, and compared
to an elastodynamic generalization of the semi-classical Gutzwiller unstable
periodic orbits formulas. In contrast to quantum mechanics, complex periodic
orbits associated with the surface Rayleigh waves dominate the low-frequency
spectrum, and already the two-cavity system displays chaotic features.Comment: 7 pages, 5 eps figures, latex (with epl.cls
Two novel approaches to the content analysis of school mathematics textbooks
The analysis of the content of school textbooks, particularly in a time of cross-cultural borrowing, is a growing field restricted by the tools currently available. In this paper, drawing on the analyses of three English year-one mathematics textbooks, we show how two approaches to the analysis of sequential data not only supplement conventional frequency analyses but highlight trends in the content of such textbooks hidden from frequency analyses alone. The first, moving averages, is conventionally used in science to eliminate noise and demonstrate trends in data. The second, Lorenz curves, is typically found in the social sciences to compare different forms of social phenomena. Both, as we show, extend the range of questions that can be meaningfully asked of textbooks. Finally, we speculate as to how both approaches can be used with other forms of ordered classroom data
Small Disks and Semiclassical Resonances
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 order
of the disk radius we recover the usual semi-classical approximation; however,
for wave lengths 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.Comment: Final published version - some changes in the discussion and the
labels on one figure are correcte
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.
Periodic orbit quantization of the Sinai billiard in the small scatterer limit
We consider the semiclassical quantization of the Sinai billiard for disk
radii R small compared to the wave length 2 pi/k. 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 KKR determinant, which we generalize here for the A_1
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
c=J_0(kR)/Y_0(kR). Surprisingly, this can take any value between plus and minus
infinity within the range of validity of the diffractive approximation kR <<4.
We study the statistics exhibited by spectra for fixed values of c. It is
Poissonian for |c|=infinity, provided the disk is placed inside a rectangle
whose sides obeys some constraints. For c=0 we find a good agreement of the
level spacing distribution with GOE, whereas the form factor and two-point
correlation function are similar but exhibit larger deviations. By varying the
parameter c from 0 to infinity the level statistics interpolates smoothly
between these limiting cases.Comment: 17 pages LaTeX, 5 postscript figures, submitted to J. Phys. A: Math.
Ge
Geometrical theory of diffraction and spectral statistics
We investigate the influence of diffraction on the statistics of energy
levels in quantum systems with a chaotic classical limit. By applying the
geometrical theory of diffraction we show that diffraction on singularities of
the potential can lead to modifications in semiclassical approximations for
spectral statistics that persist in the semiclassical limit . This
result is obtained by deriving a classical sum rule for trajectories that
connect two points in coordinate space.Comment: 14 pages, no figure, to appear in J. Phys.
Spectral statistics in chaotic systems with a point interaction
We consider quantum systems with a chaotic classical limit that are perturbed
by a point-like scatterer. The spectral form factor K(tau) for these systems is
evaluated semiclassically in terms of periodic and diffractive orbits. It is
shown for order tau^2 and tau^3 that off-diagonal contributions to the form
factor which involve diffractive orbits cancel exactly the diagonal
contributions from diffractive orbits, implying that the perturbation by the
scatterer does not change the spectral statistic. We further show that
parametric spectral statistics for these systems are universal for small
changes of the strength of the scatterer.Comment: LaTeX, 21 pages, 7 figures, small corrections, new references adde
Periodic Orbit Quantization beyond Semiclassics
A quantum generalization of the semiclassical theory of Gutzwiller is given.
The new formulation leads to systematic orbit-by-orbit inclusion of higher
contributions to the spectral determinant. We apply the theory to
billiard systems, and compare the periodic orbit quantization including the
first contribution to the exact quantum mechanical results.Comment: revte
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