On the boundary conditions in self-adjoint multi-interval Sturm–Liouville problems

In this paper, we completely solve a problem whose two-interval case has been studied recently by Wang et al. in [A. Wang, J. Sun, A. Zettl, Two-interval Sturm–Liouville operators in modified Hilbert spaces, J. Math. Anal. Appl. 328 (2007) 390–399]. The problem is to give an explicit description of all boundary conditions defining the self-adjoint differential operators associated with several Sturm–Liouville equations together, i.e., the so-called multi-interval Sturm–Liouville problems. The explicit description obtained in this paper only uses hermitian matrices. As an application, we then show that a generic self-adjoint boundary condition can not come from any direct sum of self-adjoint operators associated with the individual Sturm–Liouville equations, and it depends non-trivially on the multiples used in the definition of the so-called maximum operator.