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One-dimensional perturbations of a differential operator and the inverse scattering problem. (English. Russian original) Zbl 0569.34026

Sel. Math. Sov. 4, 19-35 (1985); translation from Problems of Mechanics in mathematical physics, Moscow 1976, 279-294 (1976).
In Section 1, the author discusses the basic definitions of the special theory for first-order differential matrix operators and evaluates the general formula for the coefficients of an operator obtained from a given one-dimensional perturbation. Section 2 serves as an introduction to scattering theory for Dirac operators. The new result here is the assertion that the scattering matrix is a continuous function of the potential. In Section 3, it is proved that the general formula for a one- dimensional perturbation may be used to reduce the inverse problem for the operator, \[ L=\left( \begin{matrix} 1\\ 0\end{matrix} \begin{matrix} 0\\ - 1\end{matrix} \right)d/dx+\left( \begin{matrix} 0\\ \bar q(x)\end{matrix} \begin{matrix} q(x)\\ 0\end{matrix} \right),\quad q(x)\to 0,\quad x\to \pm \infty, \] to the simpler problem of reconstructing the operator from the scattering matrix.
Reviewer: N.L.Maria

MSC:

34L99 Ordinary differential operators
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