On degenerate Sturm-Liouville b oundary value problems on geometric graphs

Abstract

The concept of degenerate and non-degenerate boundary value problems was introducedby V.A. Marchenko. Non-degenerate boundary value problems according to the classification ofBirkhoff are divided into regular and irregular boundary conditions. This paper gives examples ofdegenerate and non-degenerate Sturm-Liouville boundary value problems with Birkhoff irregularboundary conditions on a star graph. These examples summarize the results of V.A. Sadovnichyand his co-authors, as well as the work of B.E. Kanguzhin with co-authors. For the Sturm-Liouvilleoperator with symmetrical coefficients on an interval similar effect was observed degeneration inthe works of M. Stoun. In the case of higher-order differential operators with symmetric coefficientson the interval, the degeneracy effect is indicated in V.A. Sadovnichy and B.E. Kanguzhin. Theeffect when the same Sturm-Liouville boundary value problem, depending on the properties ofthe potential, can have a discrete or continuous spectrum was previously noted in the monographby B.E. Kanguzhin and M.A. Sadybekov. The basic properties of the system of eigenfunctionsand associated functions in the space of quadratically summable functions of Birkhoff irregularboundary value Sturm-Liouville boundary value problems on a finite interval were also studiedthere.

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Published
2020-04-05
How to Cite
KANGUZHIN, B. E.; SEITOVA, А. А.. On degenerate Sturm-Liouville b oundary value problems on geometric graphs. Journal of Mathematics, Mechanics and Computer Science, [S.l.], v. 105, n. 1, p. 79-86, apr. 2020. ISSN 2617-4871. Available at: <https://bm.kaznu.kz/index.php/kaznu/article/view/711>. Date accessed: 07 june 2020.
Keywords degenerate boundary value problems, non-degenerate boundary value problems, regular and irregular boundary conditions, Sturm-Liouville boundary value problem, star graph