The Borg-Marchenko theorem with a continuous spectrum. (English) Zbl 1108.34009

Chernov, Nikolaj (ed.) et al., Recent advances in differential equations and mathematical physics. UAB international conference on differential equations and mathematical physics, Birmingham, AL, USA, March 29–April 2, 2005. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3840-7/pbk). Contemporary Mathematics 412, 15-30 (2006).
Authors’ summary: 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.
For the entire collection see [Zbl 1097.34002].


34A55 Inverse problems involving ordinary differential equations
34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
34B24 Sturm-Liouville theory
34L05 General spectral theory of ordinary differential operators
Full Text: arXiv