Completeness of the products of solutions to PDE and uniqueness theorems in inverse scattering

Abstract

Let N=(u:( Del 2+k2)u=0), N1=( psi :( Del 2+k2-q(x)) psi =0) in D contained in/implied by R3, where D is a bounded domain, k=constant >0, q(x) in Linfinity (D). Suppose that f in LP(D), p>or=1, and integral fu psi dx=0 for all u in N and psi in N1. Then f=0. Results of this type are used to prove uniqueness theorems in inverse scattering. In particular, the author proves that the scattering amplitude A( theta ', theta ,k) known at a fixed k>0 for all theta ', theta in S2 determines the compactly supported potential q(x) uniquely. He also proves that the surface data u(x,y,k), For all x, y in P=(x:x3=0) at a fixed k>0 determine the compactly supported nu (x) in L2(D) uniquely. Here ( Del 2+k2+k2 nu (x))=- delta (x-y) in R3, D contained in/implied by R-3=(x:x3<0) is a bounded domain.