Algorithmically verifiable characterization of the class of scattering amplitudes for small potentials (Q923274)

Let \(A(\theta ',\theta,k)\) be the scattering amplitude corresponding to a potential \(q\in Q(\beta):=\{q:\;q=\bar q,\quad (1+| x|)^{\beta}| q(x)| \leq c\},\beta >3\). For sufficiently small q, such that \(\sup_{k>0}k\| A\|_{L^ 2(S^ 2)\to L^ 2(S^ 2)}<2\pi\) an algorithmically verifiable necessary and sufficient condition is given for \(A(\theta ',\theta,k)\), \(\theta ',\theta \in S^ 2\), \(k\in (0,\infty)\), to be the scattering amplitude corresponding to a \(q\in Q(\beta)\). The result is based on the characterization of the class of scattering amplitudes given by the author earlier [in the book ``Inverse problems: An interdisciplinary study'', \textit{P. C. Sabatier} (ed.) (Acad. Press, New York, 1987), pp. 153-167].