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

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.


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