# Set of eigenvalues of the boundary problem

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.

• 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. May 30, 2020 at 8:48