EIGENCURVES FOR 2-PARAMETER STURM-LIOUVILLE EQUATIONS

This paper concerns two-parameter Sturm-Liouville problems of the form -(p(x)y')' + q(x)y = (lambda r(x) + mu)y, a less than or equal to les s than or equal to b with self-adjoint boundary conditions at a and b. The set of (lambda, mu) is an element of R(2) for which there exists a nontrivial y satisfying the differential equation and the boundary c onditions turns out to be a countable union of graphs of analytic func tions. Our focus is on these graphs, which are termed eigencurves in t he literature. Although eigencurves have been used in a variety of way s for about a century, they seem comparatively underdeveloped in their own right. Our plan is to give motivation for the topic, elementary p roperties of eigencurves, illustrations on a simple example first stud ied by Richardson in 1918 (and since then by several authors), and som e natural questions which may whet the reader's appetite. Some of thes e questions lead to new types of inverse Sturm-Liouville problems.