Inverse problems for the differential operator on the graph with a cycle with different orders on different edges

Authors

  • Natalia Bondarenko

DOI:

https://doi.org/10.5556/j.tkjm.46.2015.1694

Keywords:

Geometrical Graphs, Differential Operators, Inverse Spectral Problems, Weyl-type Matrices, Method of Spectral Mappings

Abstract

We consider a variable order differential operator on a graph with a cycle. We study inverse spectral problems for this operator by the system of spectra. Uniqueness theorems are proved, and constructive algorithms are obtained for the solution of the inverse problems.

Author Biography

Natalia Bondarenko

Department ofMechanics andMathematics, Saratov State University, Astrakhanskaya 83, Saratov 410012, Russia.

References

J. Langese, G. Leugering and J. Schmidt, Modelling, analysis and control of dynamic elastic multi-link structures, Birkhauser, Boston (1994).

T. Kottos and U. Smilansky, Quantum chaos on graphs, Phys. Rev. Lett., 79(1997), 4794--4797.

P. Kuchment, Quantum graphs. Some basic structures, Waves Random Media, 14 (2004), S107--S128.

M. I. Belishev, Boundary spectral inverse problem on a class of graphs (trees) by the BC method, Inverse Problems, 20(2004), 647--672.

Y. Pokornyi and A. Borovskikh, Differential equations on networks $($geometric graphs$)$, J. Math. Sci. (N.Y.) 119 (2004), 691--718.

Analysis on Graphs and Its Applications, edited by P. Exner, J.P. Keating, P. Kuchment, T. Sunada and Teplyaev, A. Proceedings of Symposia in Pure Mathematics, AMS, 77 (2008).

V. A. Yurko, Inverse spectral problems for differential operators on arbitrary compact graphs, J. Inverse and Ill-Posed Probl., 18(2010), 245--261.

V. A. Yurko, Inverse problems for differential of any order on trees, Matemat. Zametki, 83(2008), 139--152; English transl. in Math. Notes 83(2008), 125--137.

V. A. Yurko, Inverse problems on star-type graphs: differential operators of different orders on different edges, Central European J. Math., 12(2014), 483--499.

Y. V. Pokornyi, T. V. Beloglazova and K. P. Lazarev, On a Class of Ordinary Differential Equations of Various Orders on a Graph, Mat. Zametki, 73(2003), 469--470.

Y. V. Pokornyi, T. V. Beloglazova, E. V. Dikareva and T. V. Perlovskaya, Green function for a locally interacting system of ordinary equations of different orders, Matem. Zametki, 74(2003), 146--149; English transl. in Mathem. Notes 74(2003), 141--143.

B. M. Levitan, Inverse Sturm-Liouville Problems, Nauka, Moscow (1984) (Russian); English transl., VNU Sci. Press, Utrecht (1987).

G. Freiling and V. Yurko, Inverse Sturm-Liouville problems and theirapplications, Huntington, NY: Nova Science Publishers, 305 p.(2001).

V. A. Yurko, Inverse problems for non-selfadjoint quasi-periodic differential pencils}, Analysis and Math. Physics 2(2012), 215--230.

V. A. Yurko, Method of Spectral Mappings in the Inverse Problem Theory. Inverse and Ill-Posed Problems Series, Utrecht: VSP (2002).

V. A. Yurko, Inverse problems for Sturm-Liouville operators on graphs with a cycle, Operators and Matrices, 2 (2008), 543--553.

I. V. Stankevich, An inverse problem of spectral analysis for Hill's equations, Doklady Akad. Nauk SSSR, 192(1970),34--37 (Russian).

V. A. Marchenko and I. V. Ostrovskii, A characterization of the spectrum of the Hill operator, Mat. Sbornik, 97(1975), 540--606 (Russian); English transl. in Math. USSR Sbornik 26(1975), 493--554.

V. A. Marchenko, Sturm-Liouville Operators and their Applications, Naukova Dumka, Kiev (1977) (Russian); English transl., Birkhauser (1986).

J. Poschel and E. Trubowitz, Inverse Spectral Theory, New York, Academic Press, 1987.

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Published

2015-09-30

How to Cite

Bondarenko, N. (2015). Inverse problems for the differential operator on the graph with a cycle with different orders on different edges. Tamkang Journal of Mathematics, 46(3), 229–243. https://doi.org/10.5556/j.tkjm.46.2015.1694

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Papers