24,522 research outputs found
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
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
STUDIES ON ABLATION OF OBJECTS TRAVERSING AN ATMOSPHERE
Ablation-type thermal protection of objects traversing an atmosphere - earth and mar
Reflectionless Potentials and PT Symmetry
Large families of Hamiltonians that are non-Hermitian in the conventional
sense have been found to have all eigenvalues real, a fact attributed to an
unbroken PT symmetry. The corresponding quantum theories possess an
unconventional scalar product. The eigenvalues are determined by differential
equations with boundary conditions imposed in wedges in the complex plane. For
a special class of such systems, it is possible to impose the PT-symmetric
boundary conditions on the real axis, which lies on the edges of the wedges.
The PT-symmetric spectrum can then be obtained by imposing the more transparent
requirement that the potential be reflectionless.Comment: 4 Page
Low rank perturbations and the spectral statistics of pseudointegrable billiards
We present an efficient method to solve Schr\"odinger's equation for
perturbations of low rank. In particular, the method allows to calculate the
level counting function with very little numerical effort. To illustrate the
power of the method, we calculate the number variance for two pseudointegrable
quantum billiards: the barrier billiard and the right triangle billiard
(smallest angle ). In this way, we obtain precise estimates for the
level compressibility in the semiclassical (high energy) limit. In both cases,
our results confirm recent theoretical predictions, based on periodic orbit
summation.Comment: 4 page
Quasiclassical Random Matrix Theory
We directly combine ideas of the quasiclassical approximation with random
matrix theory and apply them to the study of the spectrum, in particular to the
two-level correlator. Bogomolny's transfer operator T, quasiclassically an NxN
unitary matrix, is considered to be a random matrix. Rather than rejecting all
knowledge of the system, except for its symmetry, [as with Dyson's circular
unitary ensemble], we choose an ensemble which incorporates the knowledge of
the shortest periodic orbits, the prime quasiclassical information bearing on
the spectrum. The results largely agree with expectations but contain novel
features differing from other recent theories.Comment: 4 pages, RevTex, submitted to Phys. Rev. Lett., permanent e-mail
[email protected]
Spectral Statistics and Dynamical Localization: sharp transition in a generalized Sinai billiard
We consider a Sinai billiard where the usual hard disk scatterer is replaced
by a repulsive potential with close to the
origin. Using periodic orbit theory and numerical evidence we show that its
spectral statistics tends to Poisson statistics for large energies when
, while for
it is independent of energy, but depends on . We apply the approach of
Altshuler and Levitov [Phys. Rep. {\bf 288}, 487 (1997)] to show that the
transition in the spectral statistics is accompanied by a dynamical
localization-delocalization transition. This behaviour is reminiscent of a
metal-insulator transition in disordered electronic systems.Comment: 8 pages, 2 figures, accepted for publication in Phys. Rev. Let
Semiclassical spatial correlations in chaotic wave functions
We study the spatial autocorrelation of energy eigenfunctions corresponding to classically chaotic systems in the semiclassical regime.
Our analysis is based on the Weyl-Wigner formalism for the spectral average
of , defined as the average over eigenstates within an energy window
centered at . In this framework is the Fourier
transform in momentum space of the spectral Wigner function . Our study reveals the chord structure that
inherits from the spectral Wigner function showing the interplay between the
size of the spectral average window, and the spatial separation scale. We
discuss under which conditions is it possible to define a local system
independent regime for . In doing so, we derive an expression
that bridges the existing formulae in the literature and find expressions for
valid for any separation size .Comment: 24 pages, 3 figures, submitted to PR
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
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