Inverse Spectral-Scattering Problem with Two Sets of Discrete Spectra for the Radial Schroedinger Equation

TL;DR

This paper extends inverse spectral theory for the radial Schrödinger equation, demonstrating that the potential and boundary conditions can be uniquely reconstructed from two sets of spectral data, including discrete and continuous spectra.

Contribution

It generalizes the classical two-spectrum uniqueness theorem to cases involving both discrete and continuous spectral data for the Schrödinger equation.

Findings

Unique determination of potential and boundary conditions from combined spectral data

Extension of Borg-Marchenko theorem to continuous spectrum cases

Reconstruction method for spectral data involving two boundary conditions

Abstract

The Schroedinger equation on the half line is considered with a real-valued, integrable potential having a finite first moment. It is shown that the potential and the boundary conditions are uniquely determined by the data containing the discrete eigenvalues for a boundary condition at the origin, the continuous part of the spectral measure for that boundary condition, and a subset of the discrete eigenvalues for a different boundary condition. This result extends the celebrated two-spectrum uniqueness theorem of Borg and Marchenko to the case where there is also a continuous spectrum