2005/12/01 by Tuncay Aktosun, Ricardo Weder, Aktosun, Tuncay +1
Mathematics · #Differential Equations and Boundary Problems #FOS: Physical sciences #Mathematical Physics (math-ph) #Numerical methods in inverse problems #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.math-ph/0512001
openalex publication_date 2005/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The Schrödinger 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.