Inverse wave scattering in the Laplace domain: a factorization method approach

Let $Δ_Λ\le λ_Λ$ 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 $Ω$. Let $u^Λ_{f}$ and $u^{0}_{f}$ denote the solutions of the wave equations corresponding to $Δ_Λ$ and to the free Laplacian $Δ$ respectively, with a source term $f$ concentrated at time $t=0$ (a pulse). We show that for any fixed $λ>λ_Λ\ge 0$ and any fixed $B\subset\subset{\mathbb R}^{n}\backslash\barΩ$, the obstacle $Ω$ can be reconstructed by the data $$ F^Λ_λf(x):=\int_{0}^{\infty}e^{-\sqrtλ\,t}\big(u^Λ_{f}(t,x)-u^{0}_{f}(t,x)\big)\,dt\,,\qquad x\in B\,,\ f\in L^{2}({\mathbb R}^{n})\,,\ \mbox{supp}(f)\subset B\,. $$ 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 $(-Δ_Λ+λ)^{-1}-(-Δ+λ)^{-1}$.

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