Sturm-Liouville problems with reducible boundary conditions. (English) Zbl 1126.34021

Binding, Paul A.; Browne, Patrick J.; Watson, Bruce A.

Proc. Edinb. Math. Soc., II. Ser. 49, No. 3, 593-608 (2006).

The regular Sturm-Liouville problem
\[ \tau y := -y^{\prime \prime} + q y = \lambda y\quad\text{ on }[0, 1], \lambda \in \mathbb C \]
subject to boundary conditions
\[ P_j (\lambda) y'(j) = Q_j (\lambda) y(j), j = 0, 1 \]
is studied, where \(q \in L^1 (0, 1)\), \(P_j\) and \(Q_j\) are polynomials with real coefficients. By removing all the common factors from \(P_j\) and \(Q_j\) the author introduces the corresponding “reduced problem”. Some comparison between the original and the reduced problems regarding Jordan chain structure, eigenvalue asymptotics and eigenfunction oscillation is made. In the last section, an example is given.

Reviewer: Chie-Ping Chu (Taipei)

MSC:

34B24	Sturm-Liouville theory
47E05	General theory of ordinary differential operators
34L20	Asymptotic distribution of eigenvalues, asymptotic theory of eigenfunctions for ordinary differential operators

Keywords:

eigenparameter-dependent boundary conditions; Jordan chain