Inverse scattering problem in a class of nonlocal potentials. I

Inverse scattering problem is formulated for the scalar Schrödinger 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 principal 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 metod suggested is discussed.