68 research outputs found
Stability of the inverse resonance problem on the line
In the absence of a half-bound state, a compactly supported potential of a
Schr\"odinger operator on the line is determined up to a translation by the
zeros and poles of the meropmorphically continued left (or right) reflection
coefficient. The poles are the eigenvalues and resonances, while the zeros also
are physically relevant. We prove that all compactly supported potentials
(without half-bound states) that have reflection coefficients whose zeros and
poles are \eps-close in some disk centered at the origin are also close (in a
suitable sense). In addition, we prove stability of small perturbations of the
zero potential (which has a half-bound state) from only the eigenvalues and
resonances of the perturbation.Comment: 21 page
Geodesics of electrically and magnetically charged test particles in the Reissner-Nordstr\"om space-time: analytical solutions
We present the full set of analytical solutions of the geodesic equations of
charged test particles in the Reissner-Nordstr\"om space-time in terms of the
Weierstra{\ss} , and elliptic functions. Based on the
study of the polynomials in the and equations we characterize
the motion of test particles and discuss their properties. The motion of
charged test particles in the Reissner-Nordstr\"om space-time is compared with
the motion of neutral test particles in the field of a gravitomagnetic
monopole. Electrically or magnetically charged particles in the
Reissner-Nordstr\"om space-time with magnetic or electric charges,
respectively, move on cones similar to neutral test particles in the Taub-NUT
space-times
Application of approximation theory by nonlinear manifolds in Sturm-Liouville inverse problems
We give here some negative results in Sturm-Liouville inverse theory, meaning
that we cannot approach any of the potentials with integrable derivatives
on by an -parametric analytic family better than order
of .
Next, we prove an estimation of the eigenvalues and characteristic values of
a Sturm-Liouville operator and some properties of the solution of a certain
integral equation. This allows us to deduce from [Henkin-Novikova] some
positive results about the best reconstruction formula by giving an almost
optimal formula of order of .Comment: 40 page
The determination of the apsidal angles and Bertrand's theorem
We derive an expression for the determination of the apsidal angles that
holds good for arbitrary central potentials. Then we discuss under what
conditions the apsidal angles remain independent of the mechanical energy and
angular momentum in the central force problem. As a consequence, an alternative
and non-perturbative proof of Bertrand's theorem is obtained.Comment: Latex file, one figure; submitted for publicatio
New variables of separation for particular case of the Kowalevski top
We discuss the polynomial bi-Hamiltonian structures for the Kowalevski top in
special case of zero square integral. An explicit procedure to find variables
of separation and separation relations is considered in detail.Comment: 11 pages, LaTeX with Ams font
Zero modes in a system of Aharonov-Bohm fluxes
We study zero modes of two-dimensional Pauli operators with Aharonov--Bohm
fluxes in the case when the solenoids are arranged in periodic structures like
chains or lattices. We also consider perturbations to such periodic systems
which may be infinite and irregular but they are always supposed to be
sufficiently scarce
Sturm-Liouville operators with measure-valued coefficients
We give a comprehensive treatment of Sturm-Liouville operators with
measure-valued coefficients including, a full discussion of self-adjoint
extensions and boundary conditions, resolvents, and Weyl-Titchmarsh theory. We
avoid previous technical restrictions and, at the same time, extend all results
to a larger class of operators. Our operators include classical Sturm-Liouville
operators, Lax operators arising in the treatment of the Camassa-Holm equation,
Jacobi operators, and Sturm-Liouville operators on time scales as special
cases.Comment: 58 page
Effect of transition layers on the electromagnetic properties of composites containing conducting fibres
The approach to calculating the effective dielectric and magnetic response in
bounded composite materials is developed. The method is essentially based on
the renormalisation of the dielectric matrix parameters to account for the
surface polarisation and the displacement currents at the interfaces. This
makes it possible the use of the effective medium theory developed for
unbounded materials, where the spatially-dependent local dielectric constant
and magnetic permeability are introduced. A detailed mathematical analysis is
given for a dielectric layer having conducting fibres with in-plane positions.
The surface effects are most essential at microwave frequencies in
correspondence to the resonance excitation of fibres. In thin layers (having a
thickness of the transition layer), the effective dielectric constant has a
dispersion region at much higher frequencies compared to those for unbounded
materials, exhibiting a strong dependence on the layer thickness. For the
geometry considered, the effective magnetic permeability differs slightly from
unity and corresponds to the renormalised matrix parameter. The magnetic effect
is due entirely to the existence of the surface displacement currents.Comment: PDF, 33 pages, 10 figure
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