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Inverse Scattering at Fixed Energy on Surfaces with Euclidean Ends

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Abstract

On a fixed Riemann surface (M 0, g 0) with N Euclidean ends and genus g, we show that, under a topological condition, the scattering matrix S V (λ) at frequency λ > 0 for the operator Δ+V determines the potential V if \({V\in C^{1,\alpha}(M_0)\cap e^{-\gamma d(\cdot,z_0)^j}L^\infty(M_0)}\) for all γ > 0 and for some \({j\in\{1,2\}}\) , where d(z, z 0) denotes the distance from z to a fixed point \({z_0\in M_0}\) . The topological condition is given by \({N\geq \max(2g+1,2)}\) for j = 1 and by N ≥ g + 1 if j = 2. In \({\mathbb {R}^2}\) this implies that the operator S V (λ) determines any C 1, α potential V such that \({V(z)=O(e^{-\gamma|z|^2})}\) for all γ > 0.

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Authors and Affiliations

  1. DMA, U.M.R. 8553 CNRS, Ecole Normale Supérieure, 45 rue d’Ulm, F 75230, Paris Cedex 05, France

    Colin Guillarmou

  2. Department of Mathematics, University of Helsinki, PO Box 68, 00014, Helsinki, Finland

    Mikko Salo

  3. Department of Mathematics, Stanford University, Stanford, CA, 94305, USA

    Leo Tzou

Authors
  1. Colin Guillarmou
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  2. Mikko Salo
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  3. Leo Tzou
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Correspondence to Colin Guillarmou.

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Communicated by S. Zelditch

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Guillarmou, C., Salo, M. & Tzou, L. Inverse Scattering at Fixed Energy on Surfaces with Euclidean Ends. Commun. Math. Phys. 303, 761–784 (2011). https://doi.org/10.1007/s00220-011-1224-y

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