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Matrix representations of Sturm-Liouville problems with finite spectrum. (English) Zbl 1185.34032

The authors consider a regular Sturm-Liouville problem (SLP) of the form
\[ -(py')' + q y = \lambda w y \]
on a finite intervall \([a,b]\) with real coefficients \(r := 1/p, \; q, \; w \in L^1[a,b]\) equipped with separated or coupled real selfadjoint boundary conditions.
The typical representation of a SLP by means of an unbounded operator in an infinite dimensional space is not suitable in all cases. In fact, the spectrum of the SLP consists of only finitely many eigenvalues if it is of Atkinson type, i.e., there is a partition \(a = a_0 < b_0 < a_1 < b_1 < \cdots < a_n < b_n = b\) such that \(r = 0\) on \([a_k, b_k]\) and \(w_k := \int_{a_k}^{b_k} w \; \neq 0, \; k = 0,\dots , n\) and \(q = w = 0\) on \([b_{k-1}, a_k]\) and \(r_k := \int_{b_{k-1}}^{a_k} r \; \neq 0, \; k = 1,\dots, n\). Note that for this definition the SLP is reformulated as the system
\[ u' = r v, \; v' = (q - \lambda w) u \]
(where \(u = y, \; v = py'\)). Spectral properties of such problems have already been studied in earlier papers.
In the present paper a matrix representation of a SLP of Atkinson type is presented in the form \(DX = \lambda WX\) with real \(m \times m\) matrices \(D\) and \(W\). Here \(W\) is a diagonal matrix consisting of the values \(w_k\) and \(D\) is (at least in most cases) a symmetric tridiagonal matrix consisting of combinations of the values \(1/r_k\) and \(q_k := \int_{a_k}^{b_k} q\). Moreover, in \(D\) and \(W\) there appear some additional values depending on the boundary conditions. As a second important result it turns out that a SPL of Atkinson type is equivalent (in a certain sense) to a SLP with piecewise constant coefficients. Finaly a “reverse” result is proved: Given suitable matrices \(D\) and \(W\) and selfadjoint boundary conditions then a SLP of Atkinson type is constructed representing the matrix eigenvalue problem \(DX = \lambda WX\).

MSC:

34B24 Sturm-Liouville theory
15A18 Eigenvalues, singular values, and eigenvectors
47B25 Linear symmetric and selfadjoint operators (unbounded)
34B20 Weyl theory and its generalizations for ordinary differential equations
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