ABSTRACT
Readers seeking a fuller account of properties of solutions to singular Sturm-Liouville differential equations, boundary value problems, and Green's functions will find a readable account in the sections following this introduction. The singular Sturm-Liouville problems considered in the chapter have just such a basis of solutions and the bounded solution determines the eigenfunctions of related eigenvalue problems. There are two parts of the discussion. First the basic properties of the eigenvalues and eigenfunctions related to their existence, multiplicity, orthogonality, and eigenfunction expansions are established. Second the oscillatory and approximation properties of the eigenfunctions are developed from a unified perspective based on Jentzsch's theorem, Schur's theorem, and the Kellogg conditions. A corollary of the theorem establishes that many singular Sturm-Liouville eigenvalue problems have at most a finite number of negative eigenvalues. The principal results of this chapter apply to the most important class of singular Sturm-Liouville eigenvalue problems with separated boundary conditions that occur in applications.
