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(fY, 0, k) known for all W, BE Sz and a fixed k > 0 determines the compactly supported q(x)~ L*(D) uniquely; (2) the surface data u(x,y, k) known for all x, y E P := {x: x, = 0) and a tixed k > 0 determine the compactly supported u(x) E L’(D), D c RT := {x: xj < 0) uniquely. Here [V2 + k2 + k%(x)] u(x, y, k) = -6(x-y) in R’; (3) the surface data u(x, y, k) known for all x, ye P and ail 0~ k < k,, k,zO is arbitrarily small; determine a,(x), j= 1, 2, uniquely. Here V2u + k2u + k2a,(x) + V. (a2(x) VU) = -6(x-y) in R’, a, E L2(D), a, E H*(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, yo P and all k > 0 determine v(x) and h(k) uniquely. Here [V’+ k2 + k%(x)] u = -6(x-y) h(k), V(X)E L’(D), h(k) is Fourier transform of a wavelet of compact support; (5) the conductivity U(X)E W2.2(D), U(X)> c>O, is uniquely determined by the measurements of u and uuN on dD. Here N is the outward normal to dD, D c R’ is a bounded domain with a smooth boundary dD, V. (a(x)Vu)=O in D. (6) Necessary and sufficient conditions are given for a function A(B’, 6, k), B’, 6 ES, k > 0 is fixed, to be the scattering amplitude corresponding to a local potential from a certain class.