A spectral problem occurring in description of small transverse vibrations of a star graph of Stieltjes strings is considered. At all but one pendant vertices Dirichlet conditions are imposed which mean that these vertices are clamped. One vertex (the root) can move with damping in the direction orthogonal to the equilibrium position of the strings. We describe the spectrum of such spectral problem. The corresponding inverse problem lies in recovering the values of point masses and the lengths of the intervals between the masses using the spectrum and some other parameters. We propose conditions on a sequence of complex numbers and a collection of real numbers to be the spectrum of a problem we consider and the lengths of the edges, correspondingly.

中文翻译：

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悬垂顶点处阻尼的Stieltjes弦星图的正和反谱问题

考虑了在描述Stieltjes弦星图的小横向振动时出现的频谱问题。除一个悬垂顶点外，所有条件均施加Dirichlet条件，这意味着这些顶点被夹紧。一个顶点（根）可以在与弦的平衡位置正交的方向上阻尼移动。我们描述了这种频谱问题的频谱。相应的反问题在于使用频谱和其他一些参数来恢复点质量的值以及质量之间的间隔的长度。我们提出了一系列复数和实数集合的条件，这是我们考虑的问题的频谱以及边的长度。