Multidimensional inverse problems and completeness of the products of solutions to PDE

A method is given for proving uniqueness theorems for some inverse problems. The method is based on a result on completeness of the products of solutions to PDE. As an example, the following uniqueness theorems are proved: (1) the scattering amplitude A(θ′, θ, k) known for all θ′, θ ϵ S2 and a fixed k > 0 determines the compactly supported q(x) ϵ L2(D) uniquely; (2) the surface data u(x, y, k) known for all x, y ϵ P:= {x:x3 = 0} and a fixed k > 0 determine the compactly supported ν(x) ϵ L2(D), D ⊂ R−3:= {x:x3 0 is arbitrarily small; determine aj(x), j = 1, 2, uniquely. Here ▽2u + k2u + k2a1x) + ▽ · (a2(x) ▽u) = −δ(x − y) in R3, a1 ϵ L2(D), a2 ϵ H2(D); the same conclusion holds if the surface data are known at two distinct frequencies. (4) The surface data u(x, y, k) known for all x, y ϵ P and all k > 0 determine v(x) and h(k) uniquely. Here [▽2 + k2 + k2ν(x)] u = −δ(x − y) h(k), ν(x) ϵ L2(D), h(k) is Fourier transform of a wavelet of compact support; (5) the conductivity σ(x)ϵ W2,2(D), σ(x) ⩾ c > 0, is uniquely determined by the measurements of u and σuN on ∂D. Here N is the outward normal to ∂D, D ⊂ R3 is a bounded domain with a smooth boundary ∂D, ▽ · (σ(x) ▽u) = 0 in D. (6) Necessary and sufficient conditions are given for a function A(θ′, θ, k), θ′, θ ϵ S2, k > 0 is fixed, to be the scattering amplitude corresponding to a local potential from a certain class.