Limits of Sturm–Liouville eigenvalues when the interval shrinks to an end point

We investigate the behaviour of the eigenvalues of a self-adjoint Sturm–Liouville problem with a separated boundary condition when the interval of the problem shrinks to an end point. It is shown that all the eigenvalues, except possibly the first, approach $+\infty$. The choices of the boundary condition are found for which the first eigenvalue tends to $+\infty$, independent of the coefficient functions, and the same is done for the $-\infty$ limit. For the remaining choices of the boundary condition, several types of condition on the coefficient functions are given, so that the first eigenvalue has a finite or infinite limit and, when the limit is finite, an explicit expression for the limit is obtained. Moreover, numerous examples are presented to illustrate these results, and a construction is given to perturb the finite-limit case to the no-limit case.