On inverse scattering for the Klein-Gordon equation

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.