281 research outputs found

    Inverse boundary problems for polyharmonic operators with unbounded potentials

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    We show that the knowledge of the Dirichlet-to-Neumann map on the boundary of a bounded open set in RnR^n for the perturbed polyharmonic operator (−Δ)m+q(-\Delta)^m +q with q∈Ln/2mq\in L^{n/2m}, n>2mn>2m, determines the potential qq in the set uniquely. In the course of the proof, we construct a special Green function for the polyharmonic operator and establish its mapping properties in suitable weighted L2L^2 and LpL^p spaces. The LpL^p estimates for the special Green function are derived from LpL^p Carleman estimates with linear weights for the polyharmonic operator

    Generalized backscattering and the Lax-Phillips transform

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    Using the free-space translation representation (modified Radon transform) of Lax and Phillips in odd dimensions, it is shown that the generalized backscattering transform (so outgoing angle ω=Sθ\omega =S\theta in terms of the incoming angle with SS orthogonal and \Id-S invertible) may be further restricted to give an entire, globally Fredholm, operator on appropriate Sobolev spaces of potentials with compact support. As a corollary we show that the modified backscattering map is a local isomorphism near elements of a generic set of potentials.Comment: Minor changes, typos corrected, references adde
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