A partial inverse Sturm-Liouville problem on an arbitrary graph

The Sturm-Liouville operator with singular potentials of class W2-1 on a graph of arbitrary geometrical structure is considered. We study the partial inverse problem, which consists in the recovery of the potential on a boundary edge of the graph from a subspectrum under the assumption that the potentials on the other edges are known a priori. We obtain (i) the uniqueness theorem, (ii) a reconstruction algorithm, (iii) global solvability, and (iv) local solvability and stability for this inverse problem. Our method is based on reduction of the partial inverse problem on a graph to the Sturm-Liouville problem on a finite interval with entire analytic functions in the boundary condition.