Interference Effects in Semiclassical Orbits truecm
We investigate the relationship between the power spectrum of a wave field
and that of a spatially uncorrelated source exciting it. Correlation functions
are also examined. It is found that interference effects can significantly
alter the structure of the power spectrum, leading to oscillations even
when the power spectrum of the source is smooth. We derive a semiclassical
approximation in which these oscillations are related to orbits of the
geometrical limit of the wave system. We also derive a trace formula
that approximates a spatial average of the wave power spectrum as a sum
over periodic orbits. These calculations explain the structure of a measured
power spectrum of the fluctuations in the height of a fluid surface generated
by the circular hydraulic jump, which provided the motivation for the study.
(S.C. Creagh) and P. Dimon
Periodic Orbits and Spectral Correlations of Pseudointegrable Systems
(S.C. Creagh) and P. Dimon truecm
Periodic Orbits and Spectral Correlations of Pseudointegrable Systems truecm
Properties of periodic orbit families in pseudointegrable billiards are studied and it is found that they obey the sum rule where and refer respectively to the length and area occupied by a primitive periodic orbit family. The proliferation law follows from this. Numerical investigations reveal that the average area increases initially before saturating. This is perhaps the first demonstration that asymptotically even for systems that are not almost-integrable.
It is further found that off-diagonal terms in the
spectral form factor do contribute unlike the case of intergable
dynamics. Finally, it is shown that periodic orbits give
good estimates of two-point correlations in the semiclassical
quantum spectrum. Diffraction corrections are thus more
important near the ground state.
The Classical Spectrum of Rational Polygons and Their Quantization Using
D. Biswas truecm
The Classical Spectrum of Rational Polygons and Their Quantization Using Arbitrary Trajectories truecm
The trace of the classical evolution operator, for the flow in rational polygonal billiards is studied.
By restricting to the 2-dimensional invariant surface, it is shown that the trace, where N refers to the number of billiard copies in the invariant surface, is the area occupied by a primitive periodic orbit family of length and l is the length traversed by the particle in time t with a velocity v. Using semi-classical methods is then re-expressed as where are the quantum energy eigenvalues and is a Bessel function. This result is also derived directly for integrable polygons.
Any function constructed using the phase-space variable
and averaged over the constant energy surface thus has
oscillatory contributions from quantal eigenenergies.
This is used to demonstrate that arbitrary non-periodic
trajectories can be used as an effective quantization
tool in such systems.
Topological Chaos in a Non-ergodic System
D. Biswas truecm
Topological Chaos in a Non-ergodic System truecm
An exponential proliferation of periodic orbits in a dynamical system is associated with metric chaos or the fact that trajectories that are initially close separate exponentially with time. The proliferation rate is in fact linked to a basic sum rule obeyed by periodic orbits arising from the fact that in a bounded system the particle never escapes.
A situation in which trajectories suffer mode conversion and
splitting is studied as a small wave-length limit of the
Navier equation which describes elasto-mechanical displacements.
While most geometries lead to ergodicity, it is found that for
a square, the dynamics is non-ergodic and hence non-chaotic.
It is argued however that due to an exponential decay in
intensity, the basic conservation law should lead to an
exponential proliferation of periodic orbits. This is demonstrated
numerically and this novel pheomenon is referred to as topological
chaos in a non-ergodic situation.
Intermittency and Periodic Orbit Zeta Functions
D. Biswas truecm
Intermittency and Periodic Orbit Zeta Functions truecm
The understanding of intermittency, i.e. the
coexistence of (almost) regular and chaotic behaviour in connected ergodic
components of a dynamical system is a major challenge in describing and
characterising dynamical properties of a generic system. In Hamiltonian
flows, intermittency is generally introduced through stable islands or
marginal stable fixed points. The influence of intermittency on periodic
orbit product formulas, which are suitable representations of the Fredholm
determinant or (in semiclassical approximation) of the quantum spectral
determinant, was studied. A regularisation of divergences originating from
the regular part of the dynamics good be given and could be applied
successfully to carry out the first purely semiclassical
quantisation of the resonance spectrum of Hydrogen in a strong magnetic field.
G. Tanner and (K.T. Hansen)
Entire Spectral Determinants in Semiclassical Quantization,
G. Tanner and (K.T. Hansen) truecm
Entire Spectral Determinants in Semiclassical Quantization, truecm
We investigate spectral determinants in semiclassical quantization
that are entire functions in the complex plane since they are given by
a multiplicative quasi classical evolution operator. The spectra are
being compared to the usual Gutzwiller-Voros results aswell as to the
exact quantum mechanics.
G. Tanner, P. Cvitanovic, P. Rosenqvist, (G. Vattay) and (A. Wirzba)
Fredholm Theory and the Semiclassical Approximation
G. Tanner, P. Cvitanovic, P. Rosenqvist, (G. Vattay) and (A. Wirzba) truecm
Fredholm Theory and the Semiclassical Approximation truecm
The semiclassical approximation for chaotic systems usually produces
a divergent sum over classical trajectories.
The Fredholm method is a way of combining Bogomolny's
T-operator with the old Fredholm theory for solving integral equation,
to obtain a general scheme of resummation valid for various quantities,
not necessarily the density of states. We are applying it to
individual wavefunctions, to get a resummed formula permitting to
predict the occurence of scars (enhancement of wavefunctions around
classical periodic orbits). We also plan to apply it to scattering
systems such as the three-disks system or mesoscopic systems.
B. Georgeot, (S. Fishman and R.E. Prange)
Ray-Splitting and Quantum Chaos
B. Georgeot, (S. Fishman and R.E. Prange) truecm
Ray-Splitting and Quantum Chaos truecm
Ray-splitting is a phenomenon whereby a ray incident on a boundary
splits into more than one ray travelling away from the boundary.
This corresponds to the short-wavelength limit of wave systems where
at some place (discontinuity of the potential, boundary...) a
wave is decomposed into two other (for example, one transmitted and
one reflected). We extend results and techniques from quantum chaos
to this kind of systems, and test these extensions on a simple
model. This should enable to treat afterwards various kind of
problems, in particular in accoustics.
B. Georgeot, (T.M. Antonsen, Jr., R. Bluemel, E. Ott and R.E. Prange)
Diffraction Effects in the Gutzwiller Trace Formula
B. Georgeot, (T.M. Antonsen, Jr., R. Bluemel, E. Ott and R.E. Prange) truecm
Diffraction Effects in the Gutzwiller Trace Formula truecm
By using the geometrical theory of diffraction introduced by Keller,
we have incorporated diffractive or creeping periodic orbits in the Gutzwiller
trace formula and in the related spectral determinant. The numerical
calculations has been performed on the 2- and 3-disk scatering systems
and on the confocal hyperbolae. Also point scatterers has been considered.
A new type of system with 3 disks on a line is now under investigation.
In this system the semiclassical degeneracy of the scattering resonances
is lifted purely due to inclusion of creeping periodic orbits.
P. Rosenqvist, (G. Vattay), (A. Wirzba) and (N. Whelan)
Differential Equations to Compute Corrections to the Gutzwiller Trace
P. Rosenqvist, (G. Vattay), (A. Wirzba) and (N. Whelan) truecm
Differential Equations to Compute Corrections to the Gutzwiller Trace Formula truecm
By considering local Schrödinger problems in the neighborhood
of classical periodic orbits it is possible to construct an expansion
of the local eigenvalues.
The corrections can be found successively by using ordinary differential
By defining the local spectral determinants we can
get an corrected expression for the global spectral determinant
by multiplying the corrected local determinants together. The method has
been specialized to billard systems, and a general fortran code has been
develloped that can calculate the first correction to any
two-dimensional billard system. Numerical calculations has been performed
on the 2- and 3-disk scattering systems.
(G. Vattay) and P. Rosenqvist.
(G. Vattay) and P. Rosenqvist.