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Inverse Scattering Problem with Underdetermined Data

Published online by Cambridge University Press:  17 July 2014

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Abstract

Consider the Schrödinger operator − ∇2 + q with a smooth compactly supported potential q, q = q(x),xR3.

Let A(β,α,k) be the corresponding scattering amplitude, k2 be the energy, αS2 be the incident direction, βS2 be the direction of scattered wave, S2 be the unit sphere in R3. Assume that k = k0> 0 is fixed, and α = α0 is fixed. Then the scattering data are A(β) = A(β,α0,k0) = Aq(β) is a function on S2. The following inverse scattering problem is studied: IP: Given an arbitrary fL2(S2) and an arbitrary small number ϵ> 0, can one find qC0(D) , where DR3 is an arbitrary fixed domain, such that ||Aq(β) − f(β)|| L2(S2)<ϵ? A positive answer to this question is given. A method for constructing such a q is proposed. There are infinitely many such q, not necessarily real-valued.

Type
Research Article
Copyright
© EDP Sciences, 2014

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