Volume 10, no. 3Pages 156 - 162 # Spectral Problems on Compact Graphs

S.I. Kadchenko, S.N. Kakushkin, G.A. ZakirovaThe method of finding the eigenvalues and eigenfunctions of abstract discrete semi-bounded operators on compact graphs is developed. Linear formulas allowing to calculate the eigenvalues of these operators are obtained. The eigenvalues can be calculates starting from any of their numbers, regardless of whether the eigenvalues with previous numbers are known. Formulas allow us to solve the problem of computing all the necessary points of the spectrum of discrete semibounded operators defined on geometric graphs. The method for finding the eigenfunctions is based on the Galerkin method. The problem of choosing the basis functions underlying the construction of the solution of spectral problems generated by discrete semibounded operators is considered. An algorithm to construct the basis functions is developed. A computational experiment to find the eigenvalues and eigenfunctions of the Sturm - Liouville operator defined on a two-ribbed compact graph with standard gluing conditions is performed. The results of the computational experiment showed the high efficiency of the developed methods.

Full text- Keywords
- perturbed operators; eigenvalues; eigenfunctions; compact graph; continuity conditions; Kirchhoff conditions.
- References
- 1. Bayazitova A.A. The Sturm - Liouville Problem on Geometric Graph. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2010, no. 16 (192), issue 5, pp. 4-10. (in Russian)

2. Vlasova E.A., Zarubin V.S., Kuvyrkin G.N. Priblizhennyye metody matematicheskoy fiziki [Approximate Methods of Mathematical Physics]. Moscow, Bauman MSTU, 2004. 704 p.

3. Kadchenko S.I., Kakushkin S.N. The Numerical Methods of Eigenvalues and Eigenfunctions of Perturbed Self-Adjoin Operator Finding. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2012, no. 27 (286), issue 13, pp. 45-57. (in Russian)

4. Kadchenko S.I. Numerical Method for the Solution of Inverse Problems Generated by Perturbations of Self-Adjoint Operators by Method of Regularized Traces. Vestnik of Samara State University. Natural Science Series, 2013, no. 6 (107), pp. 23-30. (in Russian)