Discretization of inverse scattering on a half line

Abstract

We solve inverse scattering problem for Schrödinger operators with compactly supported potentials on the half line. We discretize S-matrix: we take the value of the Smatrix on some infinite sequence of positive real numbers. Using this sequence obtained from S-matrix we recover uniquely the potential by a new explicit formula, without the Gelfand-Levitan-Marchenko equation.Deducated to Sergei Kuksin (Paris and Moscow) on the occasion of his 65-th birthday . Here n o (f ) is the multiplicity of 0 as a zero of a function f and note that n o (ψ) 1.

Eigenvalues of Schrödinger operators on finite and infinite intervals

“…3) Jost and Kohn [15] adapted the method of Gelfand and Levitan to determine a potential (exponentially decaying) of a Schrödinger operator by the spectral data. Theorem 1.4 is used in the paper [22] to show that Jost and Kohn solution is not complete and there exists another solution, which gives a compactly supported potential.…”