298 research outputs found
Periodic Orbits and Spectral Statistics of Pseudointegrable Billiards
We demonstrate for a generic pseudointegrable billiard that the number of
periodic orbit families with length less than increases as , where is a constant and is the average area occupied by these families. We also find that
increases with before saturating. Finally, we show
that periodic orbits provide a good estimate of spectral correlations in the
corresponding quantum spectrum and thus conclude that diffraction effects are
not as significant in such studies.Comment: 13 pages in RevTex including 5 figure
Semiclassical Inequivalence of Polygonalized Billiards
Polygonalization of any smooth billiard boundary can be carried out in
several ways. We show here that the semiclassical description depends on the
polygonalization process and the results can be inequivalent. We also establish
that generalized tangent-polygons are closest to the corresponding smooth
billiard and for de Broglie wavelengths larger than the average length of the
edges, the two are semiclassically equivalent.Comment: revtex, 4 ps figure
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
Scale Anomaly and Quantum Chaos in the Billiards with Pointlike Scatterers
We argue that the random-matrix like energy spectra found in pseudointegrable
billiards with pointlike scatterers are related to the quantum violation of
scale invariance of classical analogue system. It is shown that the behavior of
the running coupling constant explains the key characteristics of the level
statistics of pseudointegrable billiards.Comment: 10 pages, RevTex file, uuencode
Level statistics and eigenfunctions of pseudointegrable systems: dependence on energy and genus number
We study the level statistics (second half moment and rigidity
) and the eigenfunctions of pseudointegrable systems with rough
boundaries of different genus numbers . We find that the levels form energy
intervals with a characteristic behavior of the level statistics and the
eigenfunctions in each interval. At low enough energies, the boundary roughness
is not resolved and accordingly, the eigenfunctions are quite regular functions
and the level statistics shows Poisson-like behavior. At higher energies, the
level statistics of most systems moves from Poisson-like towards Wigner-like
behavior with increasing . Investigating the wavefunctions, we find many
chaotic functions that can be described as a random superposition of regular
wavefunctions. The amplitude distribution of these chaotic functions
was found to be Gaussian with the typical value of the localization volume
. For systems with periodic boundaries we find
several additional energy regimes, where is relatively close to the
Poisson-limit. In these regimes, the eigenfunctions are either regular or
localized functions, where is close to the distribution of a sine or
cosine function in the first case and strongly peaked in the second case. Also
an interesting intermediate case between chaotic and localized eigenfunctions
appears
Level spacing distribution of pseudointegrable billiard
In this paper, we examine the level spacing distribution of the
rectangular billiard with a single point-like scatterer, which is known as
pseudointegrable. It is shown that the observed is a new type, which is
quite different from the previous conclusion. Even in the strong coupling
limit, the Poisson-like behavior rather than Wigner-like is seen for ,
although the level repulsion still remains in the small region. The
difference from the previous works is analyzed in detail.Comment: 11 pages, REVTeX file, 3 PostScript Figure
Kinetic detection of osmium(VI) ester intermediates during the OsO4‐mediated aqueous dihydroxylation of chloroethylenes
The kinetics and mechanism of the cis dihydroxylation of cis‐1,2‐ dichloroethylene, trans‐1,2‐dichloroethylene, and trichloroethylene by osmium tetroxide was studied systematically in aqueous solution. The stoichiometry of the process was determined based on the principle of continuous variation of reactant ratios with spectrophotometric detection. The results always showed 1:1 stoichiometry, which is in agreement with dihydroxylation. All three reactions were found to proceed in two distinct steps. The first step occurred on a time scale of seconds and was associated with a minor change in absorbance and was identified as the formation of a 1:1 adduct between the two reagents, which is the osmium(VI) ester that plays a decisive role in catalytic applications. This species is formed in an equilibrium that is very much shifted toward the reactants, so the osmium(VI) complex is a short‐lived intermediate of the process, which is detected kinetically, but its concentration is never high enough for structural characterization. The second reaction is accompanied by major spectral changes; it involves the formation of the final products. Our results clearly show that it is possible to detect the intermediate of the process by careful kinetic studies. It is also possible that the same strategy might be successful in other OsO4‐dependent dihydroxylation processes
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
Complex Periodic Orbits and Tunnelling in Chaotic Potentials
We derive a trace formula for the splitting-weighted density of states
suitable for chaotic potentials with isolated symmetric wells. This formula is
based on complex orbits which tunnel through classically forbidden barriers.
The theory is applicable whenever the tunnelling is dominated by isolated
orbits, a situation which applies to chaotic systems but also to certain
near-integrable ones. It is used to analyse a specific two-dimensional
potential with chaotic dynamics. Mean behaviour of the splittings is predicted
by an orbit with imaginary action. Oscillations around this mean are obtained
from a collection of related orbits whose actions have nonzero real part
Evanescent wave approach to diffractive phenomena in convex billiards with corners
What we are going to call in this paper "diffractive phenomena" in billiards
is far from being deeply understood. These are sorts of singularities that, for
example, some kind of corners introduce in the energy eigenfunctions. In this
paper we use the well-known scaling quantization procedure to study them. We
show how the scaling method can be applied to convex billiards with corners,
taking into account the strong diffraction at them and the techniques needed to
solve their Helmholtz equation. As an example we study a classically
pseudointegrable billiard, the truncated triangle. Then we focus our attention
on the spectral behavior. A numerical study of the statistical properties of
high-lying energy levels is carried out. It is found that all computed
statistical quantities are roughly described by the so-called semi-Poisson
statistics, but it is not clear whether the semi-Poisson statistics is the
correct one in the semiclassical limit.Comment: 7 pages, 8 figure
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