Inverse spectral problem for the matrix Sturm-Liouville operator

We will consider the self-adjoint matrix Sturm-Liouville operator on a finite interval with singular potential of class $W_2^{-1}$ and with the general self-adjoint boundary conditions. This operator generalizes the Sturm-Liouville operators on geometrical graphs. We will discuss the inverse problem that consists in the recovery of the considered operator from the spectral data (eigenvalues and weight matrices). For this inverse problem, the uniqueness theorem has been proved, the necessary and sufficient conditions of solvability and a reconstruction procedure have been obtained. The problem solution is based on the method of spectral mappings. In addition, we will consider the application of the main results to the Sturm-Liouville operator on a star-shaped graph.