Consider the following two eigenvalue problems: (0.1)

\begin{cases}\label{eqn:1abs}y"(x)+[\lambda^2-q(x)]y(x)=0, 0 \leq x \leq \pi,\\[3pt] y(0)=0, ay'(\pi)+\lambda y(\pi)=0, \end{cases}

(0.2)

\begin{cases} z"(x)+[\mu^2-q(x)]z(x)=0, 0 \leq x \leq \pi,\\[3pt] z'(0)=0, az'(\pi)+\mu z(\pi)=0, \end{cases}

where $q(x)$ is real-valued and integrable on [0, $\pi$ ]. Let $\{\lambda_n\}_{n\in \mathbb{Z}\setminus \{0\}}$ and $\{\mu_n\}_{n\in \mathbb{Z}\setminus \{0\}}$ denote the eigenvalues of equations (0.1) and (0.2), respectively. Then

\[\cdots\lt\mu_{-3}\lt\lambda_{-2}\lt\mu_{-2}\lt\lambda_{-1}\lt\mu_{-1}\lt\mu_1\lt\lambda_1\lt\mu_2\lt\lambda_2\lt\mu_3\lt\cdots.\]

$\lambda_n$

$\mu_n$

$\pi$

$|n|-1$

In this paper, an extremal eigenvalue problem to the Sturm-Liouville equations with discontinuous coefficients and volume constraint is investigated. Liouville transformation is applied to change the problem into an equivalent minimization problem. Finite element method is proposed and the convergence for the finite element solution is established. A monotonic decreasing algorithm is presented to solve the extremal eigenvalue problem. A global convergence for the algorithm in the continuous case is proved. A few numerical results are given to depict the efficiency of the method.

Form domains are characterized for regular $2n$-th order differential equations subject to general self-adjoint boundary conditions depending affinely on the eigenparameter. Corresponding modes of convergence for eigenfunction expansions are studied, including uniform convergence of the first $n\,-\,1$ derivatives.