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 and Imaging, 2021, 15 (2) : 257-270. doi: 10.3934/ipi.2020063
References:
[1]

O. BoykoO. 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. 

[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.

[3]

G. Gladwell, Inverse Problems in Vibration, Kluwer Academic Publishers, Dordrecht, 2004.

[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.

[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.

[6]

V. A. Marchenko, Introduction to The Theory of Inverse Problems of Spectral Analysis (in Russian), Acta, Kharkov, 2005.

[7]

O. MartynyukV. 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.

[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.

[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.

[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.

[11]

V. PivovarchikN. 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.

[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.

[13]

K. Veselić, On linear vibrational systems with one dimensional damping, Appl. Anal., 29 (1988), 1-18.  doi: 10.1080/00036818808839770.

[14]

K. Veselić, On linear vibrational systems with one dimensional damping II, Integr. Equ. Oper. Theory, 13 (1990), 883-897.  doi: 10.1007/BF01198923.

show all references

References:
[1]

O. BoykoO. 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. 

[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.

[3]

G. Gladwell, Inverse Problems in Vibration, Kluwer Academic Publishers, Dordrecht, 2004.

[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.

[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.

[6]

V. A. Marchenko, Introduction to The Theory of Inverse Problems of Spectral Analysis (in Russian), Acta, Kharkov, 2005.

[7]

O. MartynyukV. 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.

[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.

[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.

[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.

[11]

V. PivovarchikN. 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.

[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.

[13]

K. Veselić, On linear vibrational systems with one dimensional damping, Appl. Anal., 29 (1988), 1-18.  doi: 10.1080/00036818808839770.

[14]

K. Veselić, On linear vibrational systems with one dimensional damping II, Integr. Equ. Oper. Theory, 13 (1990), 883-897.  doi: 10.1007/BF01198923.

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