Inverse problems for higher order differential systems with regular singularities on star-type graphs

Main Article Content

Vjacheslav Yurko

Abstract

We study an inverse spectral problem for arbitrary order ordinary differential equations on compact star-type graphs when differential equations have regular singularities at boundary vertices. As the main spectral characteristics we introduce and study the so-called Weyl-type matrices which are generalizations of the Weyl function (m-function) for the classical Sturm-Liouville operator. We provide a procedure for constructing the solution of the inverse problem and prove its uniqueness.

Article Details

How to Cite
Yurko, V. (2015). Inverse problems for higher order differential systems with regular singularities on star-type graphs. Tamkang Journal of Mathematics, 46(3), 257–268. https://doi.org/10.5556/j.tkjm.46.2015.1754
Section
Papers
Author Biography

Vjacheslav Yurko

Department ofMathematics, Saratov State University, Astrakhanskaya 83, Saratov 410012, Russia.

References

E. Montrol, Quantum theory on a network, J. Math. Phys., 11, no.2(1970), 635--648.

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.

Yu. Pokornyi and A. Borovskikh, Differential equations on networks (geometric graphs), J. Math. Sci. (N.Y.), 119, no.6(2004), 691--718.

Yu. Pokornyi and V. Pryadiev, The qualitative Sturm-Liouville theory on spatial networks, J. Math. Sci. (N.Y.), 119 (2004), 788--835.

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

V.A. Yurko, Inverse spectral problems for Sturm-Liouville operators on graphs,

Inverse Problems, 21(2005), 1075--1086.

B. M. Brown and R. Weikard, A Borg-Levinson theorem for trees, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 461, no.2062 (2005), 3231--3243.

V. A. Yurko, Inverse problems for Sturm-Liouville operators on bush-type graphs}, Inverse Problems, 25,(2009), 105008, 14pp.

V. A. Yurko, An inverse problem for Sturm-Liouville operators on A-graphs, Applied Math. Lett., 23,(2010), 875--879.

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

C.-F. Yang, Inverse spectral problems for Sturm-Liouville operators on a d-star graph,J. Math. Anal. Appl., 365(2010), 742--749.

V. A. Yurko, An inverse problem for higher-order differential operators on star-type graphs, Inverse Problems, 23, no.3(2007), 893--903.

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. Marchenko, Sturm--Liouville operators and their applications, ``Naukova Dumka", Kiev, 1977;English transl., Birkhauser, 1986.

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

K. Chadan, D. Colton, L. Paivarinta and W. Rundell, An introduction to inverse scattering and inverse spectral problems, SIAM Monographs on Mathematical Modeling and Computation. SIAM, Philadelphia, PA, 1997.

G. Freiling and V. A. Yurko, Inverse Sturm-Liouville Problems and their Applications, NOVA Science Publishers, New York, 2001.

R. Beals, P. Deift and C. Tomei, Direct and Inverse Scattering on the Line,Math. Surveys and Monographs, v.28. Amer. Math. Soc. Providence: RI, 1988.

V. A. Yurko, Inverse Spectral Problems for Differential Operators and their Applications, Gordon and Breach, Amsterdam, 2000.

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

V. A. Yurko, Inverse problem for differential equations with a singularity,Differ. Uravneniya, 28, no.8 (1992), 1355--1362 (Russian); Englishtransl. in Diff. Equations, 28(1992), 1100--1107.

V. A. Yurko, On higher-order differential operators with a singularity, Matem. Sbornik, 186, no.6 (1995), 133--160 (Russian); English transl. in Sbornik; Mathematics,186, no.6(1995), 901--928.

V. A. Yurko, Inverse spectral problems for higher--order differential operators with a singularity, Journal of Inverse and Ill--Posed Problems, 10(2002), 413--425.

V. A. Yurko, Higher-order differential equations having a singularity in an interior point, Results in Mathematics, 42(2002), 177--191.

B.M. Levitan and I.S. Sargsyan, Introduction to Spectral Theory, AMS Transl. of Math. Monogr. 39, Providence, 1975.