1,579 research outputs found
Uniqueness theorem for inverse scattering problem with non-overdetermined data
Let be real-valued compactly supported sufficiently smooth function,
, . It is proved that the
scattering data ,
determine uniquely. here is the scattering amplitude,
corresponding to the potential
Uniqueness of the solution to inverse scattering problem with scattering data at a fixed direction of the incident wave
Let be real-valued compactly supported sufficiently smooth function.
It is proved that the scattering data , determine uniquely. Here is a fixed
direction of the incident plane wave
Creating materials with a desired refraction coefficient
A method is given for creating material with a desired refraction
coefficient. The method consists of embedding into a material with known
refraction coefficient many small particles of size . The number of
particles per unit volume around any point is prescribed, the distance between
neighboring particles is as ,
is a fixed parameter. The total number of the embedded particle is
. The physical properties of the particles are described by
the boundary impedance of the particle,
as . The refraction coefficient is the
coefficient in the wave equation
Example of two different potentials which have practically the same fixed-energy phase shifts
It is shown that the Newton-Sabatier procedure for inverting the fixed-energy
phase shifts for a potential is not an inversion method but a parameter-fitting
procedure. Theoretically there is no guarantee that this procedure is
applicable to the given set of the phase shifts, if it is applicable, there is
no guaran- tee that the potential it produces generates the phase shifts from
which it was reconstructed. Moreover, no generic potential, specifically, no
potential which is not analytic in a neighborhood of the positive real semiaxis
can be reconstructed by the Newton-Sabatier procedure.
A numerical method is given for finding spherically symmetric compactly
supported potentials which produce practically the same set of fixed-energy
phase shifts for all values of angular momentum. Concrete example of such
potentials is given
Creating desired potentials by embedding small inhomogeneities
The governing equation is in . It is shown
that any desired potential , vanishing outside a bounded domain , can
be obtained if one embeds into D many small scatterers , vanishing
outside balls , such that in ,
outside , , . It is proved that if the number of
small scatterers in any subdomain is defined as
and is given by the formula
as , where
, then the limit of the function , does exist and solves the equation in
, where ,and . The total number of small
inhomogeneities is equal to and is of the order as .
A similar result is derived in the one-dimensional case
Heat transfer in a complex medium
The heat equation is considered in the complex medium consisting of many
small bodies (particles) embedded in a given material. On the surfaces of the
small bodies an impedance boundary condition is imposed. An equation for the
limiting field is derived when the characteristic size of the small bodies
tends to zero, their total number tends to infinity at a
suitable rate, and the distance between neighboring small bodies
tends to zero: , . No periodicity is
assumed about the distribution of the small bodies. These results are basic for
a method of creating a medium in which heat signals are transmitted along a
given line. The technical part for this method is based on an inverse problem
of finding potential with prescribed eigenvalues.Comment: arXiv admin note: text overlap with arXiv:1207.056
On deconvolution problems: numerical aspects
An optimal algorithm is described for solving the deconvolution problem of
the form given the noisy data ,
The idea of the method consists of the
representation , where is a compact operator, is
injective, is the identity operator, is not boundedly invertible, and
an optimal regularizer is constructed for . The optimal regularizer is
constructed using the results of the paper MR 40#5130.Comment: 7 figure
Recovery of a quarkonium system from experimental data
For confining potentials of the form q(r)=r+p(r), where p(r) decays rapidly
and is smooth for r>0, it is proved that q(r) can be uniquely recovered from
the data {E_j,s_j}, where E_j are the bound states energies and s_j are the
values of u'_j(0), and u_j(r) are the normalized eigenfunctions of the problem
-u_j" +q(r)u_j=E_ju_j, r>0, u_j(0)=0, ||u_j||=1, where the norm is L^2(0,
\infty) norm. An algorithm is given for recovery of p(r) from few experimental
data
Electromagnetic Wave Scattering by Small Impedance Particles of an Arbitrary Shape
Scattering of electromagnetic (EM) waves by one and many small ()
impedance particles of an arbitrary shape, embedded in a homogeneous
medium, is studied. Analytic formula for the field, scattered by one particle,
is derived. The scattered field is of the order , where
is a number. This field is much larger than in the
Rayleigh-type scattering. An equation is derived for the effective EM field
scattered by many small impedance particles distributed in a bounded domain.
Novel physical effects in this domain are described and discussed
Spectral properties of Schr\"{o}dinger-type operators and large-time behavior of the solutions to the corresponding wave equation
Let be a linear, closed, densely defined in a Hilbert space operator, not
necessarily selfadjoint.
Consider the corresponding wave equations
&(1) \quad \ddot{w}+ Lw=0, \quad w(0)=0,\quad \dot{w}(0)=f, \quad
\dot{w}=\frac{dw}{dt}, \quad f \in H.
&(2) \quad \ddot{u}+Lu=f e^{-ikt}, \quad u(0)=0, \quad \dot{u}(0)=0, where
is a constant. Necessary and sufficient conditions are given for the
operator not to have eigenvalues in the half-plane Re and not to have
a positive eigenvalue at a given point . These conditions are given
in terms of the large-time behavior of the solutions to problem (1) for generic
.
Sufficient conditions are given for the validity of a version of the limiting
amplitude principle for the operator .
A relation between the limiting amplitude principle and the limiting
absorption principle is established
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