Recovering Sturm–Liouville Operators on a Graph from Pairwise Disjoint Spectra

In this paper, we study an inverse spectral problem for the Sturm-Liouville equation on a three-star graph with the Neumann and Dirichlet boundary conditions in the boundary vertices and matching conditions in the internal vertex. As spectral characteristics, we consider the spectrum of the main problem together with the spectra of two Neumann-Dirichlet problems and one Dirichlet-Dirichlet problem on the edges of the graph and investigate their properties and asymptotic behavior. We prove that if these four spectra do not intersect, then the inverse problem of recovering the potential is uniquely solvable . We give an algorithm for the construction of the potential corresponding to this quadruple of spectra.