Inverse scattering problem in a class of nonlocal potentials. 1

Inverse scattering problem is formulated for the scalar Schroedinger equation on the semi-axis in a family of phase - equivalent (nonlocal, in general case) potentials. A new method of solving this problem is suggested which satisfies the solvability, unambiguity and constructivity conditions. Initial assumptions of the method are essentially based on physically general conditions of two-particle unitarity, orthogonality and completeness of the wave functions. It is shown that in the case of scattering data corresponding to the Riemann-Hilbert problem solvable in the class of rational functions, the principle integral equation of the method is reduced on a dense subclass of separable finite rank potentials to a system of algebraic second order equations. Extension of the method to the relativistic case is carried out. A number of related problems exactly solvable by the method suggested is discussed