On characteristic functions of equilateral regular star-trees

Author N. A. Rozhenko
Odessa National I.I.Mechnikov University, University of Brunei Darussalam

Abstract A spectral problem generated by the Sturm-Liouville equation on the edges of a regular equilateral finite star-tree with the Dirichlet boundary conditions at the pendant vertices and continuity and Kirchhoff's conditions at the interior vertices is considered. The potential in the Sturm-Liouville equations on the edges is the same on each edge and symmetric with respect to the edge midpoint. The structure of the function whose set of zeros coincides with the spectrum of such a problem (characteristic function) is described.
Keywords eigenvalue; characteristic function; multiplicity; recurrent relations; Dirichlet boundary conditions; Kirchhoff conditions
Reference 1. G. Borg, Eine Umkehrung der Sturm-Liouvilleschen Eigenwertaufgabe, Acta Math., 78 (1946), 1–96.

2. R. Carlson, Adjoint and self-adjoint differential operators on graphs, Electron. J. Diff. Eqns., 6 (1998), 10 p.

3. R. Carlson, Hill’s equation on a homogeneous tree, Electron. J. Diff. Eqns., 23 (1997), 1–30.

4. C. Cattaneo, The spectrum of the continuous Laplacian on a graph, Mh. Math., 124 (1997), 215–235.

5. S. Currie, B. Watson, Eigenvalue asymptotics for differential operators on graphs, J. Comp. Appl. Math., 182 (2005), 13–31.

6. J. Friedman, J.-P. Tillich, Wave equations for graphs and the edge-based Laplacian, Pacific J. Math., 216 (2004), ¹2, 229–266.

7. P. Exner, Weakly coupled states on branching graphs, Lett. Math. Phys., 38 (1996), ¹3, 313–320.

8. P. Exner, A duality between Schrodinger operators on graphs and certain Jacobi matrices, Annales de l’I. H. P., section A, 66 (1997), ¹4, 359–371.

9. I. Kac, V. Pivovarchik, On multiplicity of a quantum graph spectrum, J. Phys. A: Math. Theor., 44 (2011), 14 p.

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

11. P. Kuchment, Quantum graphs: an introduction and a breif survey, ’Analysis on Graphs and its Applications’ Proc. Symp. Pure Math. AMS, (2008), 291–314.

12. P. Kuchment, Quantum graphs II: Some spectral properties of quantum and combinatorial graphs, J. Phys. A: Math., Gen., 38 (2005), 4887–4900.

13. V. Kostrikin, R. Schrader, Kirchhoff’s rule for quantum wires, J. Phys. A: Math. Gen., 32 (1999), 595–630.

14. C.-K. Law, V. Pivovarchik, Characteristic functions of quantum graphs, J. Phys. A: Math. Theor., 42 (2009), 11 p.

15. B. Levitan, Inverse Sturm-Liouville Problems, VSP, Zeist Nauka Moscow, 1987. (in Russian)

16. B. Levitan, M. Gasymov, Determination of a differential equation by two of its spectra, Uspekhi Mat. Nauk, 19 (1964), ¹2(116), 3–63. (in Russian)

17. V. Marchenko, Sturm-Liouville operators and applications, Naukova Dumka, Kiev (1977), English translation: Oper. Theory Adv. Appl., V.22, Birkh.auser Verlag, Basel, 1986.

18. K. Pankrashkin, Spectra of Schr.odinger operators on equilateral quantum graphs, Lett. Math. Phys., 77 (2006), ¹2, 139–154.

19. V. Pivovarchik, Inverse problem for the Sturm-Liouville equation on a simple graph, SIAM J. Math. Anal., 32 (2000), 801–819.

20. V. Pivovarchik, Inverse problem for the Sturm-Liouville equation on a star-shaped graph, Math. Nachr., 13–14 (2007), 1595–1619.

21. V. Pivovarchik, N. Rozhenko, Inverse Sturm-Liouville problem on equilateral regular tree, (2011) to apear in Appl. Analysis.

22. Yu. Pokornyi, V. Pryadiev, The qualitative Sturm-Liouville theory on spatial networks, J. Mathematical Sciences, 119 (2004), ¹6, 788–835.

23. S. Seshu, M. Reed, Linear graphs and electrical networks, Addison-Wesley Pub. Co., 1961.

24. M. Solomyak, On the spectrum of the Laplacian on regular metric trees, Waves Random Media, 14 (2004), 155–171.

25. C. Texier, G. Montambaux, Scattering theory on graphs, J. Phys. A: Math. Gen., 34 (2001), 10307–10326.

26. C. Texier, On the spectrum of the Laplace operator of metric graphs attached at a vertex – spectral determinant approach, J. Phys. A: Math. Theor., 41 (2008), 085207.

Pages 65-75
Volume 37
Issue 1
Year 2012
Journal Matematychni Studii
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