Titlebook: Inverse Problems in Quantum Scattering Theory; Khosrow Chadan,Pierre C. Sabatier Book 19771st edition Springer Science+Business Media New

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,The Gel’fand-Levitan-Jost-Kohn Method, the radial Schrödinger equation, is that of Gel’fand and Levitan, and Jost and Kohn, and is the subject of the present chapter. The second method is that of Marchenko, which uses the Jost solution. It will be studied in the following chapters. Each one has its own advantages, and permits us to clar

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,Applications of Gel’fand — Levitan Equation,ional cases, the most general case solvable in closed form is when the kernel is degenerate. In this case, it is known that one can solve the equation algebraically. The most interesting instances of this situation are the following: In the first one, we start from a potential .. which has no bound

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The Marchenko Method,phic in Im . 0, and satisfies there, for . ≠ 0, the bound .. It follows from this that . − ..is an .. function of . on any line parallel to the real axis, and that.We know also that .(.,.), and therefore .(., .), are continuous on the real k-axis.

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Examples,. In fact, in the third section of chapter IV we studied an even more general case consisting of an initial arbitrary potential for which everything is known, and an increment which has the effect of multiplying the initial Jost function by a rational function of .

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Special Classes of Potentials,roblem for them can be dealt with either by the Gel’fand-Levitan or by the Marchenko method, a special technique has been found by Martin which permits solving the direct as well as the inverse problem by an iterative method based on special properties of the Laplace transform. We follow his method.

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Nonlocal Separable Interactions,e two-body Schrödinger equation is easily solvable for them, and leads to closed expressions for a large class of such interactions. They have also been used very systematically with Faddeev equations for the three-body problem. Their main feature is that the partial .-matrix has a very simple form,

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Miscellaneous Approaches to the Inverse Problems at Fixed ,,ore complicated problems. The study is limited to radial problems (for “one dimensional” problems see Chapter XVII). Throughout this chapter, most results are stated without proof and the reader who is interested in these proofs should refer to the literature. The notations are either the ones that

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