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Recovering asymptotics of Coulomb-like potentials from fixed energy scattering data. (English) Zbl 0927.58016

It is known that a scattering metric on the manifold with boundary \((X,\partial X)\) can be written in the form \(g=x^{-4}dx^2+x^{-2}h\) for some boundary defining function \(x\), with \(h\) a symmetric tensor restricting to a positive definite form on \(T(\partial X)\). A long range potential is then a potential in the class \(C^\infty (X)\) and a Coulomb-like potential is a long range potential of the form \(Ax+O(x^2)\) for some \(A\in \mathbb{R}\). R. B. Melrose [Lect. Notes Pure Appl. Math. 161, 85-130 (1994; Zbl 0837.35107)] associated a scattering matrix \(\lambda \neq 0\longrightarrow S(\lambda)\) to such metrics and potentials. In the special case where \(A=0\), it was shown by R. B. Melrose and M. Zworski [Invent. Math. 124, 389-436 (1996; Zbl 0855.58058)] that \(S(\lambda)\) is a classical Fourier integral operator of order \(0\).
In this paper, the author proves that a similar result holds when \(A\) is nonzero and the fact that the asymptotics of Coulomb-like potentials can be recovered from the scattering matrix for various manifolds including Euclidean space.

MSC:

58J50 Spectral problems; spectral geometry; scattering theory on manifolds
58J37 Perturbations of PDEs on manifolds; asymptotics
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