A Catalogue of Sturm-Liouville Differential Equations

Abstract

The idea for this catalogue follows from the conference entitled: Bicentenaire de Charles François Sturm held at the University of Geneva, Switzerland from 15 to 19 September 2003. One of the main interests for this meeting involved the historical development of the theory of Sturm-Liouville differential equations. This theory began with the original work of Sturm from 1829 to 1836 and was then followed by the short but significant joint paper of Sturm and Liouville in 1837, on second-order linear ordinary differential equations with an eigenvalue parameter. The details of the early development of Sturm-Liouville theory, from the beginnings about 1830, are given in a historical survey paper of Jesper Lützen (1984), in which paper a complete set of references may be found to the relevant work of both Sturm and Liouville. The catalogue commences with sections devoted to a brief summary of Sturm-Liouville theory including some details of differential expressions and equations, Hilbert function spaces, differential operators, classification of interval endpoints, boundary condition functions and the Liouville transform. There follows a collection of more than 50 examples of Sturm-Liouville differential equations; many of these examples are connected with well-known special functions, and with problems in mathematical physics and applied mathematics. For most of these examples the interval endpoints are classified within the relevant Hilbert function space, and boundary condition functions are given to determine the domains of the relevant differential operators. In many cases the spectra of these operators are given. The author is indebted to many colleagues who have responded to re-quests for examples and who checked successive drafts of the catalogue.