15,955 research outputs found
Quantum Spectra of Triangular Billiards on the Sphere
We study the quantal energy spectrum of triangular billiards on a spherical
surface. Group theory yields analytical results for tiling billiards while the
generic case is treated numerically. We find that the statistical properties of
the spectra do not follow the standard random matrix results and their peculiar
behaviour can be related to the corresponding classical phase space structure.Comment: 18 pages, 5 eps figure
Spectral fluctuations and 1/f noise in the order-chaos transition regime
Level fluctuations in quantum system have been used to characterize quantum
chaos using random matrix models. Recently time series methods were used to
relate level fluctuations to the classical dynamics in the regular and chaotic
limit. In this we show that the spectrum of the system undergoing order to
chaos transition displays a characteristic noise and is
correlated with the classical chaos in the system. We demonstrate this using a
smooth potential and a time-dependent system modeled by Gaussian and circular
ensembles respectively of random matrix theory. We show the effect of short
periodic orbits on these fluctuation measures.Comment: 4 pages, 5 figures. Modified version. To appear in Phys. Rev. Let
Point perturbations of circle billiards
The spectral statistics of the circular billiard with a point-scatterer is
investigated. In the semiclassical limit, the spectrum is demonstrated to be
composed of two uncorrelated level sequences. The first corresponds to states
for which the scatterer is located in the classically forbidden region and its
energy levels are not affected by the scatterer in the semiclassical limit
while the second sequence contains the levels which are affected by the
point-scatterer. The nearest neighbor spacing distribution which results from
the superposition of these sequences is calculated analytically within some
approximation and good agreement with the distribution that was computed
numerically is found.Comment: 9 pages, 2 figure
The Statistics of the Points Where Nodal Lines Intersect a Reference Curve
We study the intersection points of a fixed planar curve with the
nodal set of a translationally invariant and isotropic Gaussian random field
\Psi(\bi{r}) and the zeros of its normal derivative across the curve. The
intersection points form a discrete random process which is the object of this
study. The field probability distribution function is completely specified by
the correlation G(|\bi{r}-\bi{r}'|) = .
Given an arbitrary G(|\bi{r}-\bi{r}'|), we compute the two point
correlation function of the point process on the line, and derive other
statistical measures (repulsion, rigidity) which characterize the short and
long range correlations of the intersection points. We use these statistical
measures to quantitatively characterize the complex patterns displayed by
various kinds of nodal networks. We apply these statistics in particular to
nodal patterns of random waves and of eigenfunctions of chaotic billiards. Of
special interest is the observation that for monochromatic random waves, the
number variance of the intersections with long straight segments grows like , as opposed to the linear growth predicted by the percolation model,
which was successfully used to predict other long range nodal properties of
that field.Comment: 33 pages, 13 figures, 1 tabl
Deviations from Berry--Robnik Distribution Caused by Spectral Accumulation
By extending the Berry--Robnik approach for the nearly integrable quantum
systems,\cite{[1]} we propose one possible scenario of the energy level spacing
distribution that deviates from the Berry--Robnik distribution. The result
described in this paper implies that deviations from the Berry--Robnik
distribution would arise when energy level components show strong accumulation,
and otherwise, the level spacing distribution agrees with the Berry--Robnik
distribution.Comment: 4 page
Whirling Waves and the Aharonov-Bohm Effect for Relativistic Spinning Particles
The formulation of Berry for the Aharonov-Bohm effect is generalized to the
relativistic regime. Then, the problem of finding the self-adjoint extensions
of the (2+1)-dimensional Dirac Hamiltonian, in an Aharonov-Bohm background
potential, is solved in a novel way. The same treatment also solves the problem
of finding the self-adjoint extensions of the Dirac Hamiltonian in a background
Aharonov-Casher
Semi-classical calculations of the two-point correlation form factor for diffractive systems
The computation of the two-point correlation form factor K(t) is performed
for a rectangular billiard with a small size impurity inside for both periodic
or Dirichlet boundary conditions. It is demonstrated that all terms of
perturbation expansion of this form factor in powers of t can be computed
directly by semiclassical trace formula. The main part of the calculation is
the summation of non-diagonal terms in the cross product of classical orbits.
When the diffraction coefficient is a constant our results coincide with
expansion of exact expressions ontained by a different method.Comment: 42 pages, 10 figures, Late
A Trace Formula for Products of Diagonal Matrix Elements in Chaotic Systems
We derive a trace formula for , where
is the diagonal matrix element of the operator in the energy basis
of a chaotic system. The result takes the form of a smooth term plus
periodic-orbit corrections; each orbit is weighted by the usual Gutzwiller
factor times , where is the average of the classical
observable along the periodic orbit . This structure for the orbit
corrections was previously proposed by Main and Wunner (chao-dyn/9904040) on
the basis of numerical evidence.Comment: 8 pages; analysis made more rigorous in the revised versio
A Class of Parameter Dependent Commuting Matrices
We present a novel class of real symmetric matrices in arbitrary dimension
, linearly dependent on a parameter . The matrix elements satisfy a set
of nontrivial constraints that arise from asking for commutation of pairs of
such matrices for all , and an intuitive sufficiency condition for the
solvability of certain linear equations that arise therefrom. This class of
matrices generically violate the Wigner von Neumann non crossing rule, and is
argued to be intimately connected with finite dimensional Hamiltonians of
quantum integrable systems.Comment: Latex, Added References, Typos correcte
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.
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