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Reconstruction of singularities from full scattering data by new estimates of bilinear Fourier multipliers

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Abstract

We prove that the singularities of a complex valued potential q in the Schrödinger hamiltonian Δ+q can be reconstructed from the linear Born approximation for full scattering data by averaging in the extra variables. We prove that, with this procedure, the accuracy in the reconstruction improves the previously known accuracy obtained from fixed angle or backscattering data. In particular, for \({q \in W^{\alpha,2}}\) for α ≥ 0, in 2D we recover the main singularity of q with an accuracy of one derivative; in 3D the accuracy is \({\epsilon > 1/2}\), increasing with α. This gives a mathematical basis for diffraction tomography. The proof is based on some new estimates for multidimensional bilinear Fourier multipliers of independent interest.

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Authors and Affiliations

  1. ETSI de Caminos, Universidad Politécnica de Madrid, 28040, Madrid, Spain

    J. A. Barceló

  2. Departamento de Matemáticas, Universidad Autonoma de Madrid, 28049, Madrid, Spain

    D. Faraco, A. Ruiz & A. Vargas

Authors
  1. J. A. Barceló
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  2. D. Faraco
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  3. A. Ruiz
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  4. A. Vargas
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Corresponding author

Correspondence to D. Faraco.

Additional information

J. Barceló, D. Faraco and A. Ruiz were partially supported by the project MMM2005-07652-C02-01 (MICINN, Spain), A. Vargas was supported MTM2007-60952 (MICINN, Spain), D. Faraco and A. Ruiz were partially supported by project Simumat (CAM, Spain).

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Barceló, J.A., Faraco, D., Ruiz, A. et al. Reconstruction of singularities from full scattering data by new estimates of bilinear Fourier multipliers. Math. Ann. 346, 505–544 (2010). https://doi.org/10.1007/s00208-009-0398-5

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  • DOI: https://doi.org/10.1007/s00208-009-0398-5

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