Stability of the Inverse Problem in Scattering Theory

We consider discrete self-adjoint Dirac systems determined by the potentials (sequences) {C k } such that the matrices C k are positive definite and j-unitary, where j is a diagonal m × m matrix and has m 1 entries 1 and m 2 entries −1 (m 1 + m 2 = m) on the main diagonal. We construct systems with rational Weyl functions and explicitly solve inverse problem to recover systems from the contractive rational Weyl functions. Moreover, we study the stability of this procedure. The matrices C k (in the potentials) are so called Halmos extensions of the Verblunsky-type coefficients ρ k . We show that in the case of the contractive rational Weyl functions the coefficients ρ k tend to zero and the matrices C k tend to the indentity matrix I m . MSC(2010): 34B20, 39A12, 39A30, 47A57Keywords. Discrete self-adjoint Dirac system, Weyl function, inverse problem, explicit solution, stability of solving inverse problem, asymptotics of the potential, Verblunsky-type coefficient.

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“…We note that various important early results on the stability of solving inverse problems were obtained by V.A. Marchenko (see, e.g., [13]).…”

Section: Introductionmentioning

confidence: 75%

The Discrete Self-Adjoint Dirac Systems of General Type: Explicit Solutions of Direct and Inverse Problems, Asymptotics of Verblunsky-Type Coefficients and the Stability of Solving of the Inverse Problem

Roitberg¹,

Sakhnovich²

2018

Z. mat. fiz. anal. geom.

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“…Notice that the considered inverse problem is, in general, illposed: small deviations in reflection matrix may lead to large differences in reconstructed functions. Nevertheless, our inversion procedure becomes stable when a priori restrictions are imposed on the class of functions to be reconstructed (cf [1,27]). Let D B be the class of material parameters such that sup 0<z<l |β(z)| N, where β relates to ε, µ, and κ via ( 9), ( 13) and (18).…”

Section: Discussionmentioning

confidence: 99%

Direct and inverse scattering problem for a stratified nonreciprocal chiral medium

Monvel

Shepelsky

1997

Inverse Problems

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“…(See [8,9] for stability analyses of the Cauchy problem for the KdV equation by means of scattering transforms.) It should also be mentioned that stability estimates for potentials directly in terms of their scattering data have been studied in [14,12] (see also [2,6]). …”

Section: Introductionmentioning

confidence: 98%