Inverse problem for the Sturm--Liouville equation on a simple graph (Q2706213)

A Sturm-Liouville problem on a graph of three joint strings with the free ends fixed is considered. The direct problem of the location of the spectrum and the inverse problem of recovering the potential from the spectrum are studied. First, an auxiliary problem with two strings is investigated, which leads to a quadratic operator pencil, and properties of its spectrum are discussed. This problem is studied in detail by the author [Asymptotic Anal. 26, No. 3-4, 219-238 (2001; Zbl 0999.34021)]. The spectrum of the problem on the graph is compared with the spectra on the three rays with Dirichlet boundary conditions. If these four spectra do not intersect, then the potentials on the rays are uniquely determined by them. More precisely, it is shown that the spectra have a certain asymptotic distribution. If conversely sequences of numbers satisfying this distribution are given, then, under some additional conditions, there are potentials such that the given sequences represent the corresponding spectra.