First, as a motivation, we will consider a simple Sturm-Liouville eigenvalue problem of second order with separated boundary conditions, and perform a suitable discretization to find a discrete version of the problem. We then will explicitly compute the eigenvalues of both problems and show that the eigenvalues of the discretized problem converge in a certain sense (as the norm of the partition tends to zero) to the eigenvalues of the original problem. Next, we look at problems with fourth order Sturm-Liouville equations and try to find a similar result.

Both problems can be rewritten as special cases of a linear Hamiltonian system. So we will consider such general Hamiltonian eigenvalue problems, discretize them in an appropriate way, and then present a result on the asymptotic behavior of those eigenvalues. The technique of the proof of our main result is as follows: The eigenvalues of both the continuous and the discrete problems turn out to be precisely the zeros of certain analytic functions. This enables us to employ a well-known result of Hurwitz on the zeros of a sequence of analytic functions that converges normally to another nontrivial function.

The applications of our result are as follows: Sometimes it may be hard to numerically compute the eigenvalues of a particular continuous problem. In such a case, our suggested discretization may be performed, and it may actually be easier to compute the eigenvalues of the discretized problems explicitly. Then, via our asymptotics, one may finally obtain the desired eigenvalues of the original problem.