Inverse Scattering Problem with Underdetermined Data
Mathematics Department, Kansas State
University, Manhattan, KS
Corresponding author. E-mail: firstname.lastname@example.org
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.
Mathematics Subject Classification: 35R30 / 35J10 / 81Q05
Key words: underdetermined data / inverse scattering / fixed energy / fixed incident direction
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