On the Marchenko inverse scattering procedure with partial information on the potential

Abstract

For the Schrödinger operator H = −∂2x + q(x) on L2(−∞, ∞) with a real potential q(x) from a broad class of non-decaying functions, we extend the well-known Marchenko inverse scattering method to recover q(x) from a suitably defined reflection coefficient R and the knowledge of q(x) on (0, ∞). Our main tool is the Titchmarsh–Weyl m-function associated with the Dirichlet boundary condition at x = 0.