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Inverse Scattering Problem with Underdetermined Data

Published online by Cambridge University Press:  17 July 2014

A. G. Ramm
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Abstract

Consider the Schrödinger operator − ∇2 + q with a smooth compactly supported potential q, q = q(x),x ∈ R3.

Let A(β,α,k) be the corresponding scattering amplitude, k2 be the energy, α ∈ S2 be the incident direction, β ∈ S2 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 f ∈ L2(S2) and an arbitrary small number ϵ> 0, can one find q ∈ C0∞(D) , where D ∈ R3 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.

Keywords

underdetermined datainverse scatteringfixed energyfixed incident direction
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 9 , Issue 5: Spectral problems , 2014 , pp. 244 - 253
Copyright
© EDP Sciences, 2014

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