Eigencurves for Two-Parameter Sturm-Liouville Equations

Abstract

This paper concerns two-parameter Sturm-Liouville problems of the form \[ - (p(x)y')' + q(x)y = (\lambda r(x) + \mu )y,\quad a \leqslant x \leqslant b\]with self-adjoint boundary conditions at a and b. The set of $(\lambda ,\mu ) \in {\bf R}^2 $ for which there exists a nontrivial y satisfying the differential equation and the boundary conditions turns out to be a countable union of graphs of analytic functions. Our focus is on these graphs, which are termed eigencurves in the literature. Although eigencurves have been used in a variety of ways for about a century, they seem comparatively underdeveloped in their own right. Our plan is to give motivation for the topic, elementary properties of eigencurves, illustrations on a simple example first studied by Richardson in 1918 (and since then by several authors), and some natural questions which may whet the reader's appetite. Some of these questions lead to new types of inverse Sturm-Liouville problems.