Concise Inverse Spectral Theory for Indefinite Strings

Definitions and mathematical statement

Terminology and unresolved scope. An inverse spectral problem asks for the recovery of a mathematical object from specified spectral data. In this record, the objects are called indefinite strings, but neither their admissible class nor their spectral data are defined. The record does not specify the governing equation, underlying interval, coefficient or measure hypotheses, endpoint conditions, normalization, or equivalence under which recovery is intended.

Supplied problem statement.

Find an equally concise analogue of the solution of the inverse spectral problem for indefinite strings.

The comparison implicit in "equally concise" and the particular solution to be used as a model are not identified. The supplied background says that concise complete inverse spectral results exist for important generalized string classes, including an indefinite analogue of Krein theory, while the "exact real-measure class" still lacks a comparable full characterization. That class is not defined, and its relation to the quoted request is not made explicit. Consequently, the supplied material does not determine a unique mathematical formulation with specified domains, boundary conditions, and quantifiers. These qualifications remain unresolved; no particular string model or reconstruction theorem is asserted here.

Short English statement

The supplied request seeks a concise inverse spectral theory for indefinite strings. Its background points to an unspecified real-measure class, but the string model, spectral data, endpoint conditions, and comparison result are missing. A precise mathematical target therefore requires clarification from the source. The supplied status is partially solved; this is a dated curation label, not an independently verified conclusion.

Sources

• Local registry: data/unsolvedmath/problems.json, permanent UnsolvedMath ID $30002759$, source problem code catalogue entry, title Concise Inverse Spectral Theory for Indefinite Strings; collection: Oberwolfach Reports Open Problems. The supplied proposed year is 2015; no proposer is identified.
• Supplied report citation: Spectral Theory and Weyl Functions (2015), Oberwolfach Reports, pp. 5--86, DOI 10.4171/owr/2015/1. https://doi.org/10.4171/owr/2015/1. The record describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions; no problem-specific page is supplied.
• Supplied literature source: Eckhardt and Kostenko, inverse spectral problem for indefinite strings. https://arxiv.org/abs/1409.0139.
• Supplied literature source: Eckhardt, inverse problem for star graphs of Krein strings. https://arxiv.org/abs/1305.2486.
• Supplied literature source labeled "original OWR report": https://doi.org/10.4171/owr/2015/1.
• Literature assessment dated exactly 2026-08-21: supplied status partially solved. The supplied assessment reports concise complete results for important generalized string classes, including an indefinite analogue of Krein theory, but says that the exact real-measure class lacks a comparable full characterization. This is supplied dated curation derived from the OWR material, not an independently verified literature assessment; its time-sensitive status and cited claims require independent verification.