# American Institute of Mathematical Sciences

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  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, doi: 10.3934/ipi.2020063
##### References:
 [1] O. Boyko, O. Martynyuk and V. Pivovarchik, On maximal multiplicity of eigenvalues of finite-dimensional spectral problem on a graph, Methods of Funct. Anal. Topology, 25 (2019), 104-117.   Google Scholar [2] J. Genin and J. S. Maybee, Mechanical vibrations trees, J. Math. Anal. Appl., 45 (1974), 746-763.  doi: 10.1016/0022-247X(74)90065-1.  Google Scholar [3] G. Gladwell, Inverse Problems in Vibration, Kluwer Academic Publishers, Dordrecht, 2004.  Google Scholar [4] G. Gladwell, Matrix inverse eigenvalue problems, Dynamical Inverse Problems: Theory and Application, CISM Courses and Lect., SpringerWienNewYork, Vienna, 529 (2011), 1–28. doi: 10.1007/978-3-7091-0696-9_1.  Google Scholar [5] F. R. Gantmakher and M. G. Krein, Oscillating Matrices and Kernels and Small Vibrations of Mechanical Systems (in Russian), GITTL, Moscow-Leningrad, (1950), Revised edition, AMS Chelsea Publishing, Providence, RI, 2002. doi: 10.1090/chel/345.  Google Scholar [6] V. A. Marchenko, Introduction to The Theory of Inverse Problems of Spectral Analysis (in Russian), Acta, Kharkov, 2005. Google Scholar [7] O. Martynyuk, V. Pivovarchik and C. Tretter, Inverse problem for a damped Stieltjes string from parts of spectra, Appl. Anal., 94 (2015), 2605-2619.  doi: 10.1080/00036811.2014.996874.  Google Scholar [8] M. Möller and V. Pivovarchik, Damped star graphs of Stieltjes strings, Proc. Amer. Math. Soc., 145 (2017), 1717-1728.  doi: 10.1090/proc/13367.  Google Scholar [9] M. Möller and V. Pivovarchik, Spectral Theory of Operator Pencils, Hermite-Biehler Functions, and Their Applications, Birkhäuser, Cham, 2015. doi: 10.1007/978-3-319-17070-1.  Google Scholar [10] V. Pivovarchik, Existence of a tree of Stieltjes strings corresponding to two given spectra,, J. Phys. A, 42 (2009), 375213, 16 pp. doi: 10.1088/1751-8113/42/37/375213.  Google Scholar [11] V. Pivovarchik, N. Rozhenko and C. Tretter, Dirichlet-Neumann inverse spectral problem for a star graph of Stieltjes strings, Linear Algebra Appl., 439 (2013), 2263-2292.  doi: 10.1016/j.laa.2013.07.003.  Google Scholar [12] V. Pivovarchik and C. Tretter, Location and multiplicities of eigenvalues for a star graph of Stieltjes strings, J. Difference Equ. Appl., 21 (2015), 383-402.  doi: 10.1080/10236198.2014.992425.  Google Scholar [13] K. Veselić, On linear vibrational systems with one dimensional damping, Appl. Anal., 29 (1988), 1-18.  doi: 10.1080/00036818808839770.  Google Scholar [14] K. Veselić, On linear vibrational systems with one dimensional damping II, Integr. Equ. Oper. Theory, 13 (1990), 883-897.  doi: 10.1007/BF01198923.  Google Scholar

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##### References:
 [1] O. Boyko, O. Martynyuk and V. Pivovarchik, On maximal multiplicity of eigenvalues of finite-dimensional spectral problem on a graph, Methods of Funct. Anal. Topology, 25 (2019), 104-117.   Google Scholar [2] J. Genin and J. S. Maybee, Mechanical vibrations trees, J. Math. Anal. Appl., 45 (1974), 746-763.  doi: 10.1016/0022-247X(74)90065-1.  Google Scholar [3] G. Gladwell, Inverse Problems in Vibration, Kluwer Academic Publishers, Dordrecht, 2004.  Google Scholar [4] G. Gladwell, Matrix inverse eigenvalue problems, Dynamical Inverse Problems: Theory and Application, CISM Courses and Lect., SpringerWienNewYork, Vienna, 529 (2011), 1–28. doi: 10.1007/978-3-7091-0696-9_1.  Google Scholar [5] F. R. Gantmakher and M. G. Krein, Oscillating Matrices and Kernels and Small Vibrations of Mechanical Systems (in Russian), GITTL, Moscow-Leningrad, (1950), Revised edition, AMS Chelsea Publishing, Providence, RI, 2002. doi: 10.1090/chel/345.  Google Scholar [6] V. A. Marchenko, Introduction to The Theory of Inverse Problems of Spectral Analysis (in Russian), Acta, Kharkov, 2005. Google Scholar [7] O. Martynyuk, V. Pivovarchik and C. Tretter, Inverse problem for a damped Stieltjes string from parts of spectra, Appl. Anal., 94 (2015), 2605-2619.  doi: 10.1080/00036811.2014.996874.  Google Scholar [8] M. Möller and V. Pivovarchik, Damped star graphs of Stieltjes strings, Proc. Amer. Math. Soc., 145 (2017), 1717-1728.  doi: 10.1090/proc/13367.  Google Scholar [9] M. Möller and V. Pivovarchik, Spectral Theory of Operator Pencils, Hermite-Biehler Functions, and Their Applications, Birkhäuser, Cham, 2015. doi: 10.1007/978-3-319-17070-1.  Google Scholar [10] V. Pivovarchik, Existence of a tree of Stieltjes strings corresponding to two given spectra,, J. Phys. A, 42 (2009), 375213, 16 pp. doi: 10.1088/1751-8113/42/37/375213.  Google Scholar [11] V. Pivovarchik, N. Rozhenko and C. Tretter, Dirichlet-Neumann inverse spectral problem for a star graph of Stieltjes strings, Linear Algebra Appl., 439 (2013), 2263-2292.  doi: 10.1016/j.laa.2013.07.003.  Google Scholar [12] V. Pivovarchik and C. Tretter, Location and multiplicities of eigenvalues for a star graph of Stieltjes strings, J. Difference Equ. Appl., 21 (2015), 383-402.  doi: 10.1080/10236198.2014.992425.  Google Scholar [13] K. Veselić, On linear vibrational systems with one dimensional damping, Appl. Anal., 29 (1988), 1-18.  doi: 10.1080/00036818808839770.  Google Scholar [14] K. Veselić, On linear vibrational systems with one dimensional damping II, Integr. Equ. Oper. Theory, 13 (1990), 883-897.  doi: 10.1007/BF01198923.  Google Scholar
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