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Inverse wave scattering in the Laplace domain: a factorization method approach. (English) Zbl 1439.35568

Summary: Let \(\Delta_{\Lambda}\leq\lambda_{\Lambda}\) 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 \(u^{\Lambda}_f\) and \(u^0_f\) 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 \(\lambda>\lambda_{\Lambda}\geq 0\) and any fixed \(B\Subset\mathbb{R}^n\backslash\overline\Omega\), the obstacle \(\Omega\) can be reconstructed by the data \[ F^{\Lambda}_{\lambda}f(x):=\int_0^{\infty}e^{-\sqrt\lambda\,t}\big (u^{\Lambda}_f(t,x)-u^0_f(t,x)\big)\,dt, \] \[ x\in B,\, f\in L^2(\mathbb{R}^n),\, \operatorname{supp}(f)\subset B. \] 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 \((-\Delta_{\Lambda}+\lambda)^{-1}-(-\Delta +\lambda)^{-1}\).

MSC:

35R30 Inverse problems for PDEs
47A40 Scattering theory of linear operators
47B25 Linear symmetric and selfadjoint operators (unbounded)
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