# American Institute of Mathematical Sciences

April  2021, 15(2): 257-270. doi: 10.3934/ipi.2020063

## Direct and inverse spectral problems for a star graph of Stieltjes strings damped at a pendant vertex

 1 College of Mathematics and Information Science, Shaanxi Normal University, Xi'an 710062, China 2 South Ukrainian national Pedagogical University, Staroprtofrankovskaya str., 26, Odessa 65020, Ukraine

* Corresponding author: Vyacheslav Pivovarchik

Received  May 2020 Revised  August 2020 Published  April 2021 Early access  October 2020

Fund Project: The first author is supported in part by NNSF grant 11971284

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.

Citation: Lu Yang, Guangsheng Wei, Vyacheslav Pivovarchik. Direct and inverse spectral problems for a star graph of Stieltjes strings damped at a pendant vertex. Inverse Problems & Imaging, 2021, 15 (2) : 257-270. doi: 10.3934/ipi.2020063
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