On recovering differential operators on a closed set from spectra

The Sturm–Liouville differential operators on closed sets of the real line are considered. Properties of their spectral characteristics are obtained and the inverse problem of recovering the operators from their spectra is studied. An algorithm for the solution of the inverse problem is developed and the uniqueness of the solution is established. The statement and the study of inverse spectral problems essentially depend on the structure of the closed set. We consider an important subclass of closed sets when the set is a unification of a finite number of closed intervals and isolated points. In order to solve the inverse spectral problem for this class of closed sets, we develop ideas of the method of spectral mappings. We also establish and use connections between the Weyl-type functions related to different subsets of the main closed set. Using these ideas and properties we obtain a global constructive procedure for the solution of the nonlinear inverse problem considered, and we establish the uniqueness of the solution of the inverse problem.