*Math. Model. Nat. Phenom.*Vol. 9, No. 5, 2014, pp. 244-253

## Inverse Scattering Problem with Underdetermined Data

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

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

Consider the Schrödinger operator −
∇^{2} + *q* with a smooth compactly supported
potential *q*,
*q* =
*q*(*x*)*,x* ∈
**R**^{3}.

Let
*A*(*β,α,k*)
be the corresponding scattering amplitude, *k*^{2} be the energy, *α* ∈
*S*^{2} be the incident direction,
*β* ∈
*S*^{2} be the direction of scattered wave,
*S*^{2} be the unit sphere in **R**^{3}. Assume that
*k* =
*k*_{0}> 0 is fixed, and
*α* =
*α*_{0} is fixed. Then the scattering data are
*A*(*β*) =
*A*(*β,**α*_{0},*k*_{0})
= *A*_{q}(*β*)
is a function on *S*^{2}. The following inverse scattering
problem is studied: *IP: Given an arbitrary **f* ∈
*L*^{2}(*S*^{2})*
and an arbitrary small number **ϵ*> 0*, can one find *
*q* ∈ *C*_{0}^{∞}(*D*)
*, where **D* ∈
**R**^{3}* 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.

Mathematics Subject Classification: 35R30 / 35J10 / 81Q05

Key words: underdetermined data / inverse scattering / fixed energy / fixed incident direction

*© EDP Sciences, 2014*