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 n(2) does. With an appropriate and natural interpretation of a "zero," the eigenfunctions have the usual zero properties. (C) 2001 Academic Press.