AMBARZUMYAN-TYPE THEOREMS FOR THE STURM-LIOUVILLE EQUATION ON A GRAPH

In this paper we consider the inverse spectral problem of small vibrations of a graph consisting of d, d >= 2, d is an element of N, joint inhomogeneous smooth strings which can be reduced to the Sturm-Liouville boundary value problem on a graph. This problem occurs also in quantum mechanics. An analog of Ambarzumyan's theorem is proved for the case of a Sturm-Liouville problem on the compact metric graph consisting of d segments of equal length with the Neumann boundary conditions at the pendant vertices and Kirchhoff boundary conditions at the central vertex, which case is also exceptional. We also extend Ambarzumyan's theorem of a Sturm-Liouville problem to the compact metric graph with the Dirichlet boundary conditions at the pendant vertices, by imposing an additional condition on the potential functions. The proof is based on the Gelfand-Levitan equation and variational principle.