# Green’s function of the Dirichlet problem for the differential operator on a star-shaped graph at m

## DOI:

https://doi.org/10.26577/JMMCS-2019-1-613## Keywords:

oriented graph, vertices of graph, Kifchhoff condition, vibrations of elastic networks, Dirichlet problem, extension by eigenfunctions## Abstract

In this paper, we investigate a system of second-order differential equations, which is a model of oscillatory systems with a rod structure. Problems for differential operators on graphs are currently being actively studied by mathematicians and have applications in quantum mechanics, organic chemistry, nanotechnology, the theory of waveguides and other fields of natural science. A graph is a structure consisting of "abstract" segments and vertices whose adjoining to each other is described by a certain relation. To define an operator on a given graph, it is necessary to select a set of boundary vertices. Vertices that are not boundary are called internal vertices. The differential operator on a given graph is determined not only by given differential expressions on arcs, but also by conditions of the Kirchhoff type at the internal vertices of the graph. This article solved the Dirichlet problem for a differential operator on a star graph. We used the standard gluing conditions at inner vertices and Dirichlet boundary conditions at the boundary vertices. Also in this paper, the subtraction of Green's function of a differential operator on a star shaped graph is presented. Questions from spectral theory, such as the construction of the Green function and the expansion in eigenfunctions for models from connected rods, have been little studied. Spectral analysis of differential operators on graphs is the main mathematical apparatus in solving modern problems of quantum mechanics.

## References

[2] Pokornyi U.V. O spectre nekotoryh zadach na graphah [About the spectrum of some problems on the graph], Uspehi mat.nauk. 1987. - V. 42, P. 128-129.

[3] Penkin O.M. O krayevoi zadache na graphe [About boundary value problems on a graph], Differenciyalnye uravneniya 1988. - V. 24, - P. 701-703.

[4] Von Below J. Classical solvability of linear parabolic equations on networks, Differential Equation. 1988. - V.72, P. 316-337.

[5] Von Below J. Sturm-Liouville eigenvalue problems on networks, Math. Metli. Appl. Sc. 1988. - V.10, P.383-395.

[6] Lumer G. Connecting of local operators and evolution equations on network, Lect. Notes Math. 1980.- V.787 - P. 219-234.

[7] Nicaise S. Some results on spectral theory over networks, applied to nerve impuls transmission, Lect.Notes Math. 1985. - no 1771- P. 532-541.

[8] Pokornyi U.B. Differenciyalnye uravneniya na geometricheskih graphah [Differenciail equations on geometric graphs]. M.: Phizmatlit. 2004. - P. 272-274.

[9] Kanguzhin B.E. Funkciya Grina zadachi Dirihle dlya differencialnogo operatora na grafe-zvezde[Green’s function of Dirichlet problem for differential operators on a star-shaped graphs], Vestnik KazNU 2018. - P. 67-90.

[10] Bondarenko N.P. Partial inverse problems for the Sturm-Liouville operator on star-shaped graph with mixed boundary conditions, J. Inverse Ill - Posed Probl. 2018.- P.1-12

[11] Afanasev N.A., Bulot L.P. Electrotehnika i electronika[Electrotechnik and electronik], SPbGUN and P.T. 2010. P.181-183.

[12] Zavgorodnii M.G. Sopryazhennye i samosopryajennye krayvye zadachi na geometricheskom graphe [Conjugate and selfadjoint

boundary value problems on a geometric graph], Differencial equations. 2014. - V. 50, no 4- P. 446-456.

[13] Kurasov P., Stenberg F. On the inverse scattering problem on branching graphs, J. Phys. A. Math. Gen, 2002. - V. 20. - P. 647-672.

[14] Pokornyi U.V., Priadiev V.L., Al-Obeid A. Ob oscilyacionnyh svoistvah spectra kraevoi zadachi na graphe, Matem.zametki, 1996 - V.60. - P. 468-469.

[15] Pokornyi U.V., Priadiev V.L. Nekotorye problemy kachestvennoi teorii Shturma-Liuvillya na prostranstvennyh setyah [Some problems of the qualitative theory of Sturm-Liouville on spatial networks], Uspehi mat. nauk.2004. - V. 59, - P. 115-150.

[16] Znojil M. Quantum star-graph analogues of PT-symmetric square wells // Can. J. Phys. 2012 - V.90, iss 12. - P.1287-1293.

[17] Naimark M.A. Lineinye differenciyalnye operatory [Linear differential operators] - М.: Nauka. 1969. - P.526.

[18] F.Harary, Graph theory, Addison-Wesley Publishing Company. 1969. - 274 p.

[19] P. Kurasov, M. Garjiani, Quantum graphs: PT-symmetry and reflection symmetry of the spectrum.// Journal of Mathematical Physics. 2017. - V.58.

[20] M. Astudillo, P. Kurasov, M. Usman, RT -symmetric laplace operators on star graphs: Real spectrum and selfadjointness. // Adv. Math. Phys. 2015.

## Downloads

## Published

## How to Cite

*Journal of Mathematics, Mechanics and Computer Science*,

*101*(1), 14–28. https://doi.org/10.26577/JMMCS-2019-1-613