Inverse spectral problem for a star graph of Stieltjes strings with prescribed numbers of masses on the edges
Consider a spectral problem for a star graph of Stieltjes strings. At the central vertex the generalized Neumann conditions are imposed. All but one (called the root) pendant vertices of the graph are clamped. We consider two problems:
(i) with the Neumann condition at the root (the Neumann problem),
(ii) with the Dirichlet condition at the root (the Dirichlet problem). In [V. Pivovarchik, N. Rozhenko, C. Tretter, Dirichlet – Neumann inverse spectral problem for a star graph of Stieltjes strings, Linear Algebra and Appl., 439, № 8, 2263 – 2292 (2013)], the spectra of such problems were described and the corresponding inverse problem of recovering the values of masses and lengths of the intervals between them was solved by using the spectra of the two (Neumann and Dirichlet) problems. In the present paper, in contrast to the results mentioned
above we solve the inverse problem where the number of point masses on the edges is prescribed. We find necessary and sufficient conditions guaranteeing that two sequences of real numbers are the spectra of the Dirichlet and Neumann problems for a star graph with prescribed numbers of masses on the edges and prescribed lengths of edges.
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