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Sojourn times, manifolds with infinite cylindrical ends, and an inverse problem for planar waveguides

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Abstract

We prove that two particular entries in the scattering matrix for the Dirichlet Laplacian on ℝ × (−γ, γ) \( \mathcal{O} \) determine an analytic strictly convex obstacle \( \mathcal{O} \). With an additional symmetry assumption, one entry suffices. Part of the proof is an integral identity involving an entry in the scattering matrix and a distribution related to the fundamental solution of the wave equation. This identity holds for general manifolds with infinite cylindrical ends. A consequence of this is a relationship between the singularities of the Fourier transform of an entry in the scattering matrix and the sojourn times of certain geodesics.

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Authors and Affiliations

  1. Department Of Mathematics, University Of Missouri, Columbia, Missouri, 65211, USA

    T. J. Christiansen

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  1. T. J. Christiansen
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Correspondence to T. J. Christiansen.

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Partially supported by NSF grant DMS 0500267.

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Christiansen, T.J. Sojourn times, manifolds with infinite cylindrical ends, and an inverse problem for planar waveguides. J Anal Math 107, 79–106 (2009). https://doi.org/10.1007/s11854-009-0004-5

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  • DOI: https://doi.org/10.1007/s11854-009-0004-5

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