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I'm looking for the results about the set of eigenvalues of boundary problem for differential equation \begin{equation} \bigl(p(x) u'(x; \lambda) \bigr)' + q(x) u(x; \lambda) = -\lambda w(x) u(x; \lambda), \quad x \in [0, h] \end{equation} where $p(x) > 0$, $q(x) > 0$ and $w(x) < 0$, with boundary conditions \begin{equation} u'(x; \lambda)|_{0} = u'(x; \lambda)|_{h} = 0. \end{equation} It seems that there must be only finitely many positive eigenvalues of the problem. But I haven't found any book where such problem is considered.

Actually, this problem arises in the theory of electromagnetic waves propagation in plane shielded waveguides filled with inhomogeneous medium. In this case, the first equation has the form \begin{equation} \bigl(\frac{1}{\varepsilon(x)}u'(x; \lambda)\bigr)' + u(x; \lambda) = \lambda\frac{1}{\varepsilon(x)}u(x;\lambda); \end{equation}
here $\varepsilon(x) > 0$ is a continuous function that describes permittivity of the medium. So, actually I'm interested particularly of this special case.

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    $\begingroup$ Dividing the equation by $-w$ you get a Sturm-Liouville problem which can be studied in the weighted space $L^2_w$. A. Zettl has written a very extensive and detailed book on the topic. $\endgroup$ – Giorgio Metafune May 30 '20 at 8:48
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I've managed to find the necessary information in the book "Sturm-Liouville Theory" by Anton Zettl. This book was recommended to me by @Giorgio Metafune in his comment.

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  • $\begingroup$ Please consider that questions/answers here are not only for the OP to use. In its present form your answer is of very poor quality for any potential reader except yourself. $\endgroup$ – მამუკა ჯიბლაძე Jun 27 '20 at 8:56

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