Sturm–Liouville Problems with Finite Spectrum

For every positive integer n, we construct a class of regular self-adjoint and nonself- adjoint Sturm–Liouville problems with exactly n eigenvalues.These n eigenvalues can be located anywhere in the complex plane in the non-self-adjoint case and anywhere along the real line in the self-adjoint case.The latter complements the well-known general result for right-definite Sturm–Liouville problems with an infinite number of eigenvalues, which must go to infinity asymptotically like n2 does. With an appropriate and natural interpretation of a “zero,” the eigenfunctions have the usual zero properties.