Transfer functions and local spectral uniqueness for Sturm-Liouville operators, canonical systems and strings. (English) Zbl 1365.34033
The paper mostly deals with the Sturm-Liouville operator on an interval \([0,l)\), with a self-adjoint boundary condition at one end. The author shows that the transfer function introduced by M.G. Krein is a powerful tool for proving local versions of the Borg-Marchenko uniqueness theorem, namely, for establishing conditions guaranteeing that two potential coincide on some sub-interval \([0,a]\) with \(a<l\). Particularly, the use of the transfer function simplifies the proof of B. Simon’s local versions of the Borg-Marchenko theorem. The application of the transfer function to local spectral uniqueness for other equations, such as canonical systems and string equations, is also discussed.
Reviewer: Dmitry Shepelsky (Kharkov)
MSC:
| 34A55 | Inverse problems involving ordinary differential equations |
| 34B20 | Weyl theory and its generalizations for ordinary differential equations |
| 34L40 | Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.) |
| 47A11 | Local spectral properties of linear operators |
| 47B32 | Linear operators in reproducing-kernel Hilbert spaces (including de Branges, de Branges-Rovnyak, and other structured spaces) |
Keywords:
inverse problems; Sturm-Liouville operators; canonical systems; strings; local spectral uniqueness; spectral measure; transfer function; Marchenko-Borg theoremReferences:
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