On degenerate Sturm-Liouville b oundary value problems on geometric graphs - Kanguzhin, B. E.; Seitova, А. А. (Journal of Mathematics, Mechanics and Computer Science, 2020)
The concept of degenerate and non-degenerate boundary value problems was introduced by V.A. Marchenko. Non-degenerate boundary value problems according to the classification of Birkhoff are divided into regular and irregular boundary conditions. This paper gives examples of degenerate and non-degenerate Sturm-Liouville boundary value problems with Birkhoff irregular boundary conditions on a star graph. These examples summarize the results of V.A. Sadovnichy and his co-authors, as well as the work of B.E. Kanguzhin with co-authors. For the Sturm-Liouville operator with symmetrical coefficients on an interval similar effect was observed degeneration in the works of M. Stoun. In the case of higher-order differential operators with symmetric coefficients on the interval, the degeneracy effect is indicated in V.A. Sadovnichy and B.E. Kanguzhin. The effect when the same Sturm-Liouville boundary value problem, depending on the properties of the potential, can have a discrete or continuous spectrum was previously noted in the monograph by B.E. Kanguzhin and M.A. Sadybekov. The basic properties of the system of eigenfunctions and associated functions in the space of quadratically summable functions of Birkhoff irregular boundary value Sturm-Liouville boundary value problems on a finite interval were also studied there.