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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



On inverse scattering for the Klein-Gordon equation

Author: Tomas P. Schonbek
Journal: Trans. Amer. Math. Soc. 166 (1972), 101-123
MSC: Primary 47F05; Secondary 35L05
MathSciNet review: 0298476
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Abstract: A scattering operator $ S = S(V)$ is set up for the Klein-Gordon equation $ \square u = {m^2}u(m > 0)$ perturbed by a linear potential $ V = V(x)$ to $ \square u = {m^2}u + Vu$. It is found that for each $ R > 0$ there exists a constant $ c(R)$ (of order $ {R^{2 - n}}$ as $ R \to + \infty $, n = space dimension) such that if the $ {L_1}$ and the $ {L_q}$ norm of V and $ V'$ are bounded by $ c(R),V' - V$ is either nonnegative or nonpositive, and $ V' - V$ is of compact support having diameter $ \leqq R$, then $ S(V') \ne S(V)$ or $ V' = V$. Here $ q > n/2$, and $ c(R)$ may also depend on q.

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Article copyright: © Copyright 1972 American Mathematical Society

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