INVERSE WAVE SCATTERING IN THE LAPLACE DOMAIN: A FACTORIZATION METHOD APPROACH

Let Delta(Lambda) <= lambda(Lambda) be a semi-bounded self-adjoint realization of the Laplace operator with boundary conditions (Dirichlet, Neumann, semi-transparent) assigned on the Lipschitz boundary of a bounded obstacle Omega. Let u(f)(Lambda) and u(f)(0) denote the solutions of the wave equations corresponding to Delta(Lambda) and to the free Laplacian Delta, respectively, with a source term f concentrated at time t = 0 (a pulse). We show that for any fixed lambda > lambda(Lambda) >= 0 and any fixed B subset of R-n(Omega) over bar, the obstacle Omega can be reconstructed by the dataF(lambda)(Lambda)f(x) : = integral(infinity)(0) e(-root lambda t)(u(f)(Lambda)(t, x) - u(f)(0)(t, x)) dt,x is an element of B, f is an element of L-2(R-n), supp(f) subset of B.A similar result holds in the case of screens reconstruction, when the boundary conditions are assigned only on a part of the boundary. Our method exploits the factorized form of the resolvent difference (-Delta(Lambda) + lambda)(-1) - (-Delta + lambda)(-1).