282 research outputs found
On non-overdetermined inverse scattering at zero energy in three dimensions
We develop the d-bar -approach to inverse scattering at zero energy in
dimensions d>=3 of [Beals, Coifman 1985], [Henkin, Novikov 1987] and [Novikov
2002]. As a result we give, in particular, uniqueness theorem, precise
reconstruction procedure, stability estimate and approximate reconstruction for
the problem of finding a sufficiently small potential v in the Schrodinger
equation from a fixed non-overdetermined ("backscattering type") restriction h
on of the Faddeev generalized scattering amplitude h in the complex
domain at zero energy in dimension d=3. For sufficiently small potentials v we
formulate also a characterization theorem for the aforementioned restriction h
on and a new characterization theorem for the full Faddeev function h
in the complex domain at zero energy in dimension d=3. We show that the results
of the present work have direct applications to the electrical impedance
tomography via a reduction given first in [Novikov, 1987, 1988]
Formulas for phase recovering from phaseless scattering data at fixed frequency
We consider quantum and acoustic wave propagation at fixed frequency for
compactly supported scatterers in dimension . In these framework we
give explicit formulas for phase recovering from appropriate phaseless
scattering data. As a corollary, we give global uniqueness results for quantum
and acoustic inverse scattering at fixed frequency without phase information
An effectivization of the global reconstruction in the Gel'fand-Calderon inverse problem in three dimensions
By developing the d-bar approach to global "inverse scattering" at zero
energy we give a principal effectivization of the global reconstruction method
for the Gel'fand-Calderon inverse boundary value problem in three dimensions.
This work goes back to results published by the author in 1987, 1988 and
proceeds from recent progress in the d-bar approach to inverse scattering in 3D
published by the author in 2005, 2006
Phaseless inverse scattering in the one-dimensional case
We consider the one-dimensional Schr\"odinger equation with a potential
satisfying the standard assumptions of the inverse scattering theory and
supported on the half-line . For this equation at fixed positive energy
we give explicit formulas for finding the full complex valued reflection
coefficient to the left from appropriate phaseless scattering data measured on
the left, i.e. for . Using these formulas and known inverse scattering
results we obtain global uniqueness and reconstruction results for phaseless
inverse scattering in dimension
Large time asymptotics for the Grinevich-Zakharov potentials
In this article we show that the large time asymptotics for the
Grinevich-Zakharov rational solutions of the Novikov-Veselov equation at
positive energy (an analog of KdV in 2+1 dimensions) is given by a finite sum
of localized travel waves (solitons)
Faddeev eigenfunctions for point potentials in two dimensions
We present explicit formulas for the Faddeev eigenfunctions and related
generalized scattering data for point (delta-type) potentials in two
dimensions. In particular, we obtain the first explicit examples of such
eigenfunctions with contour singularity in spectral parameter at a fixed real
energy
On the reconstruction of conductivity of bordered two-dimensional surface in R^3 from electrical currents measurements on its boundary
An electrical potential U on bordered surface X (in Euclidien
three-dimensional space) with isotropic conductivity function sigma>0 satisfies
equation d(sigma d^cU)=0, where d^c is real operator associated with complex
(conforme) structure on X induced by Euclidien metric of three-dimensional
space. This paper gives exact reconstruction of conductivity function sigma on
X from Dirichlet-to-Neumann mapping (for aforementioned conductivity equation)
on the boundary of X. This paper extends to the case of the Riemann surfaces
the reconstruction schemes of R.Novikov (1988) and of A.Bukhgeim (2008) given
for the case of domains in two-dimensional Euclidien space. The paper extends
and corrects the statements of Henkin-Michel (2008), where the inverse boundary
value problem on the Riemann surfaces was firstly considered
An analog of Chang inversion formula for weighted Radon transforms in multidimensions
In this work we study weighted Radon transforms in multidimensions. We
introduce an analog of Chang approximate inversion formula for such transforms
and describe all weights for which this formula is exact. In addition, we
indicate possible tomographic applications of inversion methods for weighted
Radon transforms in 3D
A global stability estimate for the Gel'fand-Calderon inverse problem in two dimensions
We prove a global logarithmic stability estimate for the Gel'fand-Calderon
inverse problem on a two-dimensional domain
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