Continuous analogue of the Newton method in the inverse problem of scattering theory in the presence of eigenfunctions and eigenvalues

The paper considers the inverse scattering problem---the problem of determining the potential function in a second-order Schrödinger type differential equation with initial condition depending on a parameter. The potential function is assumed to be a smooth function satisfying a suitable limit condition. Besides the Schrödinger differential equation, another first-order differential equation involving a potential function as a coefficient, called the "equation of phases", is considered. For solving the inverse scattering problem in the presence of eigenvalues and eigenfunctions, a variant of the Newton method is formulated and applied. A numerical example is given.