Inverse Sturm-Liouville Problem with Nonseparated Boundary Conditions on a Geometric Graph

The inverse Sturm-Liouville problem with nonseparated boundary conditions on a star-shaped geometric graph consisting of three edges with a common vertex is studied. It is shown that the Sturm-Liouville problem with general boundary conditions cannot be reconstructed uniquely from four spectra. A class of nonseparated boundary conditions is obtained for which two uniqueness theorems for the solution of the inverse Sturm-Liouville problem are proved. In the first theorem, the data used to reconstruct the Sturm-Liouville problem are the spectrum of the boundary value problem itself and the spectra of three auxiliary problems with separated boundary conditions. In the second theorem, instead of the spectrum of the problem itself, one only deals with five of its eigenvalues. It is shown that the Sturm-Liouville problem with these nonseparated boundary conditions can be reconstructed uniquely if three spectra of auxiliary problems and five eigenvalues of the problem itself are used as the reconstruction data. Examples of unique reconstruction of potentials and boundary conditions of the Sturm-Liouville problem posed on the graph under study are given.