The Borg-Marchenko Theorem with a Continuous Spectrum

TL;DR

This paper generalizes the Borg-Marchenko uniqueness theorem for Schrödinger operators on the half line to include cases with continuous spectra, providing new methods for potential and boundary condition recovery from spectral data.

Contribution

It extends the classical uniqueness results to potentials with continuous spectra, introducing a novel spectral data set for inversion and comparison with existing methods.

Findings

Unique determination of potential and boundary conditions from combined spectral data.

Development of a new inversion method for Schrödinger operators with continuous spectrum.

Comparison with existing spectral shift function approaches.

Abstract

The Schr\"odinger equation is considered on the half line with a selfadjoint boundary condition when the potential is real valued, integrable, and has a finite first moment. It is proved that the potential and the two boundary conditions are uniquely determined by a set of spectral 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 provides a generalization of the celebrated uniqueness theorem of Borg and Marchenko using two sets of discrete spectra to the case where there is also a continuous spectrum. The proof employed yields a method to recover the potential and the two boundary conditions, and it also constructs data sets used in various inversion methods. A comparison is made with the uniqueness result of Gesztesy and Simon using Krein's spectral shift function as the inversion data.