390 research outputs found
Asymptotic behaviour of multiple scattering on infinite number of parallel demi-planes
The exact solution for the scattering of electromagnetic waves on an infinite
number of parallel demi-planes has been obtained by J.F. Carlson and A.E. Heins
in 1947 using the Wiener-Hopf method. We analyze their solution in the
semiclassical limit of small wavelength and find the asymptotic behaviour of
the reflection and transmission coefficients. The results are compared with the
ones obtained within the Kirchhoff approximation
Energy Level Quasi-Crossings: Accidental Degeneracies or Signature of Quantum Chaos?
In the field of quantum chaos, the study of energy levels plays an important
role. The aim of this review paper is to critically discuss some of the main
contributions regarding the connection between classical dynamics,
semi-classical quantization and spectral statistics of energy levels. In
particular, we analyze in detail degeneracies and quasi-crossings in the
eigenvalues of quantum Hamiltonians which are classically non-integrable.
Summary: 1. Introduction; 2. Quasi-Crossing and Chaos; 3. Molecular
Spectroscopy; 4. Nuclear Models; 4.1 Zirnbauer-Verbaashot-Weidenmuller Model;
4.2 Lipkin-Meshow-Glick Model; 5. Particle Physics and Field Theory; 6.
Conclusions.Comment: 26 pages, Latex, 9 figures, to be published in International Journal
of Modern Physics
Semiclassical Quantisation Using Diffractive Orbits
Diffraction, in the context of semiclassical mechanics, describes the manner
in which quantum mechanics smooths over discontinuities in the classical
mechanics. An important example is a billiard with sharp corners; its
semiclassical quantisation requires the inclusion of diffractive periodic
orbits in addition to classical periodic orbits. In this paper we construct the
corresponding zeta function and apply it to a scattering problem which has only
diffractive periodic orbits. We find that the resonances are accurately given
by the zeros of the diffractive zeta function.Comment: Revtex document. Submitted to PRL. Figures available on reques
The Quantum-Classical Correspondence in Polygonal Billiards
We show that wave functions in planar rational polygonal billiards (all
angles rationally related to Pi) can be expanded in a basis of quasi-stationary
and spatially regular states. Unlike the energy eigenstates, these states are
directly related to the classical invariant surfaces in the semiclassical
limit. This is illustrated for the barrier billiard. We expect that these
states are also present in integrable billiards with point scatterers or
magnetic flux lines.Comment: 8 pages, 9 figures (in reduced quality), to appear in PR
Uniform approximations for pitchfork bifurcation sequences
In non-integrable Hamiltonian systems with mixed phase space and discrete
symmetries, sequences of pitchfork bifurcations of periodic orbits pave the way
from integrability to chaos. In extending the semiclassical trace formula for
the spectral density, we develop a uniform approximation for the combined
contribution of pitchfork bifurcation pairs. For a two-dimensional double-well
potential and the familiar H\'enon-Heiles potential, we obtain very good
agreement with exact quantum-mechanical calculations. We also consider the
integrable limit of the scenario which corresponds to the bifurcation of a
torus from an isolated periodic orbit. For the separable version of the
H\'enon-Heiles system we give an analytical uniform trace formula, which also
yields the correct harmonic-oscillator SU(2) limit at low energies, and obtain
excellent agreement with the slightly coarse-grained quantum-mechanical density
of states.Comment: LaTeX, 31 pp., 18 figs. Version (v3): correction of several misprint
Numerical investigation of iso-spectral cavities built from triangles
We present computational approaches as alternatives to the recent microwave
cavity experiment by S. Sridhar and A. Kudrolli (Phys. Rev. Lett. {\bf 72},
2175 (1994)) on iso-spectral cavities built from triangles. A straightforward
proof of iso-spectrality is given based on the mode matching method. Our
results show that the experiment is accurate to 0.3% for the first 25 states.
The level statistics resemble those of GOE when the integrable part of the
spectrum is removed.Comment: 15 pages, revtex, 5 postscript figure
Distribution of Husimi Zeroes in Polygonal Billiards
The zeroes of the Husimi function provide a minimal description of individual
quantum eigenstates and their distribution is of considerable interest. We
provide here a numerical study for pseudo- integrable billiards which suggests
that the zeroes tend to diffuse over phase space in a manner reminiscent of
chaotic systems but nevertheless contain a subtle signature of
pseudo-integrability. We also find that the zeroes depend sensitively on the
position and momentum uncertainties with the classical correspondence best when
the position and momentum uncertainties are equal. Finally, short range
correlations seem to be well described by the Ginibre ensemble of complex
matrices.Comment: includes 13 ps figures; Phys. Rev. E (in press
Periodic Orbits in Polygonal Billiards
We review some properties of periodic orbit families in polygonal billiards
and discuss in particular a sum rule that they obey. In addition, we provide
algorithms to determine periodic orbit families and present numerical results
that shed new light on the proliferation law and its variation with the genus
of the invariant surface. Finally, we deal with correlations in the length
spectrum and find that long orbits display Poisson fluctuations.Comment: 30 pages (Latex) including 11 figure
Singular continuous spectra in a pseudo-integrable billiard
The pseudo-integrable barrier billiard invented by Hannay and McCraw [J.
Phys. A 23, 887 (1990)] -- rectangular billiard with line-segment barrier
placed on a symmetry axis -- is generalized. It is proven that the flow on
invariant surfaces of genus two exhibits a singular continuous spectral
component.Comment: 4 pages, 2 figure
Classical Dynamics of Anyons and the Quantum Spectrum
In this paper we show that (a) all the known exact solutions of the problem
of N-anyons in oscillator potential precisely arise from the collective degrees
of freedom, (b) the system is pseudo-integrable ala Richens and Berry. We
conclude that the exact solutions are trivial thermodynamically as well as
dynamically.Comment: 19 pages, ReVTeX, IMSc/93/0
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