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The \({\bar \partial}\)-equation in the multidimensional inverse scattering problem. (English. Russian original) Zbl 0674.35085
Russ. Math. Surv. 42, No. 3, 109-180 (1987); translation from Usp. Mat. Nauk 42, No. 3(255), 93-152 (1987).
The objective of this paper is to give a characterization of the class of scattering amplitudes, i.e., to give a necessary and sufficient condition for a function \(A(\theta',\theta,k)\), \(\theta',\theta \in S^ 2\), \(k>0\) to be the scattering amplitude corresponding to a potential q(x) from a certain class. The authors give a characterization of the so called “non-physical scattering data”, namely a certain function \(H(x,p)\), \(k\in {\mathbb{C}}^ n\setminus ({\mathbb{R}}^ n\cup \epsilon)\), \(p\in {\mathbb{R}}^ n\), which is not obtained in a physical experiment. The characterization is given in terms of a certain equation (\({\bar \partial}\) equation (3.1)) which this function solves for complex \(k\in {\mathbb{C}}^ n\) in a certain subset of \({\mathbb{C}}^ n\). The characterization is given for the \(C^{\infty}\) potentials which decay with all their derivatives faster than any power of \(| x|\) (Schwartz’s class \({\mathcal S})\) (Theorem 4.1). The potential can be found from the function \(\lim_{k\to \infty}H(k,p)=\tilde q(p)\), where \(\tilde q\) is the Fourier transform of q(x). In Theorem 4.2 a characterization of the physical data, that is, the scattering amplitude, is given for \(q\in {\mathcal S}\) \((q\in C_{\infty}^{(\infty)}({\mathbb{R}}^ n)\) in the authors’ notation) in terms of some properties of a function \(h_{\gamma}(k,\ell)\) which solves equation (1.7) in the paper. The kernel of this equation can be written if the scattering amplitude is known. These are the basic results. There are many other results mentioned in this paper. The new results due to the authors are based on the \({\bar \partial}\) approach in inverse scattering theory introduced by Beals and Coifman.
Reviewer’s remark. If the given function \(A(\theta',\theta,k)\) is not known to be a scattering amplitude there is no discussion of the basic question of solvability of equation (1.7) in the paper. Therefore the characterization of the physical data given by the authors is not constructive: no algorithmically verifiable conditions on the data \(A(\theta',\theta,k)\) are given which allow one to say that equation (1.7) has a solution with the properties formulated in Theorem 4.2. Two different characterizations of the physical scattering data were given by the reviewer in the book “Inverse problems: An interdisciplinary study”, Acad. Press, New York, 153-167 (1987), ed. P. Sabatier (*) (see also the work of O. L. Weaver and the reviewer [Inverse Probl. 3, L 49-L 52 (1987; Zbl 0657.35103)]). These characterizations are also not algorithmically verifiable. These characterizations and the characterization obtained here independently have the following logical structure: if and only if a certain integral equation, whose kernel is defined by the given function \(A(\theta ',\theta,k)\), has a solution with certain properties, then the given function \(A(\theta ',\theta,k)\) is the scattering amplitude corresponding to a potential from a certain class. Here, this class is the Schwartz class \({\mathcal S}\) and the equation is (1.7), in (*) this class is rather broad, for example, the class \[ Q:=\{q:\quad q=\bar q,\quad | q| +| \nabla q| \leq c(1+| x|)^{-a},\quad a>3,\quad c=\text{ const } >0\} \] and the equation is (30) in (*). In the works of L. Faddeev, R.Newton, M. Ablowitz and A. Nachman cited in this work no characterization of the physical data is given (that is no necessary and sufficient condition for \(A(\theta',\theta,k)\) to be the scattering amplitude corresponding to a potential from a given class), but a number of various necessary conditions are obtained for \(A(\theta',\theta,k)\) to be the scattering amplitude. A reader can see the expression “characterization of the scattering data” used in the works on inverse scattering not in the sense that necessary and sufficient conditions are given but with the meaning that some necessary conditions are obtained. The authors write that a complete characterization of the physical scattering data is obtained in their paper. The conditions in this characterization cannot be algorithmically checked (namely, it is not known what are the algorithmically verifiable properties of \(A(\theta',\theta,k)\) which imply the existence of the solution to equation (1.7) with the properties required in Theorem 4.2) and in this respect this characterization does not differ from simpler characterizations given in (*). In (*) a characterization of the fixed-energy physical data is also given. The reviewer thinks that the studies of the non-physical data (as long as they do not provide information about physical data) are of less interest than the study of the physical data.
Reviewer: A.G.Ramm

35R30 Inverse problems for PDEs
35P25 Scattering theory for PDEs
35J10 Schrödinger operator, Schrödinger equation
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