2,932 research outputs found
Partner orbits and action differences on compact factors of the hyperbolic plane. Part I: Sieber-Richter pairs
Physicists have argued that periodic orbit bunching leads to universal
spectral fluctuations for chaotic quantum systems. To establish a more detailed
mathematical understanding of this fact, it is first necessary to look more
closely at the classical side of the problem and determine orbit pairs
consisting of orbits which have similar actions. In this paper we specialize to
the geodesic flow on compact factors of the hyperbolic plane as a classical
chaotic system. We prove the existence of a periodic partner orbit for a given
periodic orbit which has a small-angle self-crossing in configuration space
which is a `2-encounter'; such configurations are called `Sieber-Richter pairs'
in the physics literature. Furthermore, we derive an estimate for the action
difference of the partners. In the second part of this paper [13], an inductive
argument is provided to deal with higher-order encounters.Comment: to appear on Nonlinearit
Classical orbit bifurcation and quantum interference in mesoscopic magnetoconductance
We study the magnetoconductance of electrons through a mesoscopic channel
with antidots. Through quantum interference effects, the conductance maxima as
functions of the magnetic field strength and the antidot radius (regulated by
the applied gate voltage) exhibit characteristic dislocations that have been
observed experimentally. Using the semiclassical periodic orbit theory, we
relate these dislocations directly to bifurcations of the leading classes of
periodic orbits.Comment: 4 pages, including 5 figures. Revised version with clarified
discussion and minor editorial change
A generic map has no absolutely continuous invariant probability measure
Let be a smooth compact manifold (maybe with boundary, maybe
disconnected) of any dimension . We consider the set of maps
which have no absolutely continuous (with respect to Lebesgue)
invariant probability measure. We show that this is a residual (dense
C^1$ topology.
In the course of the proof, we need a generalization of the usual Rokhlin
tower lemma to non-invariant measures. That result may be of independent
interest.Comment: 12 page
Semiclassical universality of parametric spectral correlations
We consider quantum systems with a chaotic classical limit that depend on an
external parameter, and study correlations between the spectra at different
parameter values. In particular, we consider the parametric spectral form
factor which depends on a scaled parameter difference . For
parameter variations that do not change the symmetry of the system we show by
using semiclassical periodic orbit expansions that the small expansion
of the form factor agrees with Random Matrix Theory for systems with and
without time reversal symmetry.Comment: 18 pages, no figure
Periodic-Orbit Bifurcations and Superdeformed Shell Structure
We have derived a semiclassical trace formula for the level density of the
three-dimensional spheroidal cavity. To overcome the divergences occurring at
bifurcations and in the spherical limit, the trace integrals over the
action-angle variables were performed using an improved stationary phase
method. The resulting semiclassical level density oscillations and
shell-correction energies are in good agreement with quantum-mechanical
results. We find that the bifurcations of some dominant short periodic orbits
lead to an enhancement of the shell structure for "superdeformed" shapes
related to those known from atomic nuclei.Comment: 4 pages including 3 figure
Significance of Ghost Orbit Bifurcations in Semiclassical Spectra
Gutzwiller's trace formula for the semiclassical density of states in a
chaotic system diverges near bifurcations of periodic orbits, where it must be
replaced with uniform approximations. It is well known that, when applying
these approximations, complex predecessors of orbits created in the bifurcation
("ghost orbits") can produce pronounced signatures in the semiclassical spectra
in the vicinity of the bifurcation. It is the purpose of this paper to
demonstrate that these ghost orbits themselves can undergo bifurcations,
resulting in complex, nongeneric bifurcation scenarios. We do so by studying an
example taken from the Diamagnetic Kepler Problem, viz. the period quadrupling
of the balloon orbit. By application of normal form theory we construct an
analytic description of the complete bifurcation scenario, which is then used
to calculate the pertinent uniform approximation. The ghost orbit bifurcation
turns out to produce signatures in the semiclassical spectrum in much the same
way as a bifurcation of real orbits would.Comment: 20 pages, 6 figures, LATEX (IOP style), submitted to J. Phys.
Exchange-coupling constants, spin density map, and Q dependence of the inelastic neutron scattering intensity in single-molecule magnets
The Q dependence of the inelastic neutron scattering (INS) intensity of
transitions within the ground-state spin multiplet of single-molecule magnets
(SMMs) is considered. For these transitions, the Q dependence is related to the
spin density map in the ground state, which in turn is governed by the
Heisenberg exchange interactions in the cluster. This provides the possibility
to infer the exchange-coupling constants from the Q dependence of the INS
transitions within the spin ground state. The potential of this strategy is
explored for the M = +-10 -> +- 9 transition within the S = 10 multiplet of the
molecule Mn12 as an example. The Q dependence is calculated for powder as well
as single-crystal Mn12 samples for various exchange-coupling situations
discussed in the literature. The results are compared to literature data on a
powder sample of Mn12 and to measurements on an oriented array of about 500
single-crystals of Mn12. The calculated Q dependence exhibits significant
variation with the exchange-coupling constants, in particular for a
single-crystal sample, but the experimental findings did not permit an
unambiguous determination. However, although challenging, suitable experiments
are within the reach of today's instruments.Comment: 11 pages, 6 figures, REVTEX4, to appear in PR
On the Accuracy of the Semiclassical Trace Formula
The semiclassical trace formula provides the basic construction from which
one derives the semiclassical approximation for the spectrum of quantum systems
which are chaotic in the classical limit. When the dimensionality of the system
increases, the mean level spacing decreases as , while the
semiclassical approximation is commonly believed to provide an accuracy of
order , independently of d. If this were true, the semiclassical trace
formula would be limited to systems in d <= 2 only. In the present work we set
about to define proper measures of the semiclassical spectral accuracy, and to
propose theoretical and numerical evidence to the effect that the semiclassical
accuracy, measured in units of the mean level spacing, depends only weakly (if
at all) on the dimensionality. Detailed and thorough numerical tests were
performed for the Sinai billiard in 2 and 3 dimensions, substantiating the
theoretical arguments.Comment: LaTeX, 31 pages, 14 figures, final version (minor changes
On the stability of periodic orbits in delay equations with large delay
We prove a necessary and sufficient criterion for the exponential stability
of periodic solutions of delay differential equations with large delay. We show
that for sufficiently large delay the Floquet spectrum near criticality is
characterized by a set of curves, which we call asymptotic continuous spectrum,
that is independent on the delay.Comment: postprint versio
Nonperiodic echoes from mushroom billiard hats
Mushroom billiards have the remarkable property to show one or more clear cut
integrable islands in one or several chaotic seas, without any fractal
boundaries. The islands correspond to orbits confined to the hats of the
mushrooms, which they share with the chaotic orbits. It is thus interesting to
ask how long a chaotic orbit will remain in the hat before returning to the
stem. This question is equivalent to the inquiry about delay times for
scattering from the hat of the mushroom into an opening where the stem should
be. For fixed angular momentum we find that no more than three different delay
times are possible. This induces striking nonperiodic structures in the delay
times that may be of importance for mesoscopic devices and should be accessible
to microwave experiments.Comment: Submitted to Phys. Rev. E without the appendi
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