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

Abstract

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.