Inverse Scattering Problem with Underdetermined Data

Mathematics Department, Kansas State University, Manhattan, KS 66506-2602, USA

^⋆ Corresponding author. E-mail: ramm@math.ksu.edu

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) = A_q(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 ||A_q(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.