124 research outputs found
Discrete Dirac system: rectangular Weyl functions, direct and inverse problems
A transfer matrix function representation of the fundamental solution of the
general-type discrete Dirac system, corresponding to rectangular Schur
coefficients and Weyl functions, is obtained. Connections with Szeg\"o
recurrence, Schur coefficients and structured matrices are treated.
Borg-Marchenko-type uniqueness theorem is derived. Inverse problems on the
interval and semiaxis are solved.Comment: Section 2 is improved in the second version: some new results on
Halmos extension are added and arguments are simplifie
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
Multiple solutions for asteroid orbits: Computational procedure and applications
We describe the Multiple Solutions Method, a one-dimensional sampling of the six-dimensional orbital confidence region that is widely applicable in the field of asteroid orbit determination. In many situations there is one predominant direction of uncertainty in an orbit determination or orbital prediction, i.e., a ``weak'' direction. The idea is to record Multiple Solutions by following this, typically curved, weak direction, or Line Of Variations (LOV). In this paper we describe the method and give new insights into the mathematics behind this tool. We pay particular attention to the problem of how to ensure that the coordinate systems are properly scaled so that the weak direction really reflects the intrinsic direction of greatest uncertainty. We also describe how the multiple solutions can be used even in the absence of a nominal orbit solution, which substantially broadens the realm of applications. There are numerous applications for multiple solutions; we discuss a few problems in asteroid orbit determination and prediction where we have had good success with the method. In particular, we show that multiple solutions can be used effectively for potential impact monitoring, preliminary orbit determination, asteroid identification, and for the recovery of lost asteroids
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
Dirac systems with locally square-integrable potentials: direct and inverse problems for the spectral functions
We solve the inverse problems to recover Dirac systems on an interval or
semiaxis from their spectral functions (matrix valued functions) for the case
of locally square-integrable potentials. Direct problems in terms of spectral
functions are treated as well. Moreover, we present necessary and sufficient
conditions on the given distribution matrix valued function to be a spectral
function of some Dirac system with a locally square-integrable potential.
Interesting connections with Paley-Wiener sampling measures appear in the case
of scalar spectral functions.Comment: This work is an important development of our earlier papers
arXiv:1401.3605 and arXiv:1611.0055
Property C for ODE and applications to inverse problems
Property C stands for completeness of the set of products of solutions to
homogeneous linear differential equations. property C is proved in various
formulations for Schr\"odinger operators. Many applications of this property to
inverse problems and inverse scattering problems are given. It is shown what
part of the fixed-energy phase shifts determines a compactly supported
potential uniquely. It is shown that the Newton-Sabatier method is not really
an inversion method but a parameter-fitting procedure and it is proved that
this procedure cannot recover a generic potential, in particular, any potential
which is not analytic in a neighborhood of the positive real axis
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