Recovering Asymptotics of Short Range Potentials |
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Authors: | M S Joshi A Sá Barreto |
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Institution: | Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, 16 Mill Lane, Cambridge CB2 1SB, England, UK. E-mail: joshi@dpmms.cam.ac.uk, UK Department of Mathematics, Purdue University, West Lafayette IN 47907, Indiana, USA.?E-mail: sabarre@math.purdue.edu, US
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Abstract: | Any compact smooth manifold with boundary admits a Riemann metric of the form near the boundary, where x is the boundary defining function and h' restricts to a Riemannian metric, h, on the boundary. Melrose has associated a scattering matrix to such a metric which was shown by he and Zworski to be a Fourier
integral operator. It is shown here that the principal symbol of the difference of the scattering matrices for two potentials
at fixed energy determines a weighted integral of the lead term of V
1 - V
2 over all geodesics on the boundary. This is used to prove that the entire Taylor series of the potential at the boundary
is determined by the scattering matrix at a non-zero fixed energy for certain manifolds including Euclidean space.
Received: 3 January 1997 / Accepted: 15 August 1997 |
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