130 research outputs found
New, Highly Accurate Propagator for the Linear and Nonlinear Schr\"odinger Equation
A propagation method for the time dependent Schr\"odinger equation was
studied leading to a general scheme of solving ode type equations. Standard
space discretization of time-dependent pde's usually results in system of ode's
of the form u_t -Gu = s where G is a operator (matrix) and u is a
time-dependent solution vector. Highly accurate methods, based on polynomial
approximation of a modified exponential evolution operator, had been developed
already for this type of problems where G is a linear, time independent matrix
and s is a constant vector. In this paper we will describe a new algorithm for
the more general case where s is a time-dependent r.h.s vector. An iterative
version of the new algorithm can be applied to the general case where G depends
on t or u. Numerical results for Schr\"odinger equation with time-dependent
potential and to non-linear Schr\"odinger equation will be presented.Comment: 14 page
Quantum statistics of atoms in microstructures
This paper proposes groove-like potential structures for the observation of
quantum information processing by trapped particles. As an illustration the
effect of quantum statistics at a 50-50 beam splitter is investigated. For
non-interacting particles we regain the results known from photon experiments,
but we have found that particle interactions destroy the perfect bosonic
correlations. Fermions avoid each other due to the exclusion principle and
hence they are far less sensitive to particle interactions. For bosons, the
behavior can be explained with simple analytic considerations which predict a
certain amount of universality. This is verified by detailed numerical
calculations.Comment: 18 pages incl. 13 figure
Quantum Geometrodynamics I: Quantum-Driven Many-Fingered Time
The classical theory of gravity predicts its own demise -- singularities. We
therefore attempt to quantize gravitation, and present here a new approach to
the quantization of gravity wherein the concept of time is derived by imposing
the constraints as expectation-value equations over the true dynamical degrees
of freedom of the gravitational field -- a representation of the underlying
anisotropy of space. This self-consistent approach leads to qualitatively
different predictions than the Dirac and the ADM quantizations, and in
addition, our theory avoids the interpretational conundrums associated with the
problem of time in quantum gravity. We briefly describe the structure of our
functional equations, and apply our quantization technique to two examples so
as to illustrate the basic ideas of our approach.Comment: 11, (No Figures), (Typeset using RevTeX
Open Problems on Central Simple Algebras
We provide a survey of past research and a list of open problems regarding
central simple algebras and the Brauer group over a field, intended both for
experts and for beginners.Comment: v2 has some small revisions to the text. Some items are re-numbered,
compared to v
Generic model of an atom laser
We present a generic model of an atom laser by including a pump and loss term
in the Gross-Pitaevskii equation. We show that there exists a threshold for the
pump above which the mean matter field assumes a non-vanishing value in
steady-state. We study the transient regime of this atom laser and find
oscillations around the stationary solution even in the presence of a loss
term. These oscillations are damped away when we introduce a position dependent
loss term. For this case we present a modified Thomas-Fermi solution that takes
into account the pump and loss. Our generic model of an atom laser is analogous
to the semi-classical theory of the laser.Comment: 15 pages, including 5 figures, submitted to Phys. Rev. A, revised
manuscript, file also available at
http://www.physik.uni-ulm.de/quan/users/kne
Endomorphism algebras of Abelian varieties with special reference to superelliptic Jacobians
This is (mostly) a survey article. We use an information about Galois properties of points of small order on an Abelian variety in order to describe its endomorphism algebra over an algebraic closure of the ground field. We discuss in detail applications to jacobians of cyclic covers of the projective line
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