Inverse problem for the Sturm-Liouville equation on a simple graph

The spectrum of small vibrations of a graph consisting of three joint smooth strings with the free ends fixed can be reduced to the Sturm Liouville boundary problem on a graph. This problem occurs also in quantum mechanics. The spectrum of such a problem is investigated in comparison with the union of spectra of Dirichlet problems on the rays of the graph. It is shown that the eigenvalues of the spectra interlace in some sense; thus an analogue of Sturm theorem is established. If the four spectra ( the spectrum of the boundary problem on the graph and the three spectra of the mentioned Dirichlet problems) do not intersect, then the inverse problem of recovering the potentials on the rays from the four spectra is uniquely solvable. The procedure of construction of the potentials is presented.