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. 



Set of eigenvalues of the boundary problem
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.


