2009/06/17 by A. G. Ramm, Ramm, A. G.
Mathematics · Physics and Astronomy · #35R30 #81U40 #FOS: Physical sciences #Mathematical Physics (math-ph) #math-ph #math.MP #msc:35R30 #msc:81U40
paper · pdf · doi:10.48550/arxiv.0906.3211
arxiv created 2009/06/21 · arxiv updated 2009/12/01
Let A(β,α,k) be the scattering amplitude corresponding to a real-valued potential which vanishes outside of a bounded domain D⊂ \R3. The unit vector α is the direction of the incident plane wave, the unit vector β is the direction of the scattered wave, k>0 is the wave number. The governing equation for the waves is [∇2+k2-q(x)]u=0 in \R3. For a suitable class of potentials it is proved that if Aq1(-β,β,k)=Aq2(-β,β,k) ∀ β∈ S2, ∀ k∈ (k0,k1), and q1, q2∈ M, then q1=q2. This is a uniqueness theorem for the solution to the inverse scattering problem with backscattering data. It is also proved for this class of potentials that if Aq1(β,α0,k)=Aq2(β,α0,k) ∀ β∈ S21, ∀ k∈ (k0,k1), and q1, q2∈ M,then q1=q2. Here S21 is an arbitrarily small open subset of S2, and |k0-k1|>0 is arbitrarily small.