No abstract

The SLEIGN2 code is based on the ideas and methods of the original SLEIGN code of 1979. The main purpose of the SLEIGN2 code is to compute eigenvalues and eigenfunctions of regular and singular self-adjoint Sturm-Liouville problems, with both separated and coupled boundary conditions, and to approximate the continuous spectrum in the singular case. The code uses some new algorithms, which we describe, and has a driver program that offers a user-friendly interface. In this paper the algorithms and their implementations are discussed, and the class of problems to which each algorithm applied is identified.

We consider the nth eigenvalue as a function on the space of self-adjoint regular Sturm Liouville problems with positive leading coefficient and weight functions. The discontinuity of the nth eigenvalue is completely characterized. Academic Press

The eigenvalues of Sturm Liouville (SL) problems depend not only continuously but smoothly on the problem. An expression for the derivative of an eigenvalue with respect to a given parameter: an endpoint, a boundary condition, a coefficient or the weight function, is found.

Given any self-adjoint realization S of a singular Sturm-Liouville (S-L) problem, it is possible to construct a sequence {S r } of regular S-L problems with the properties (i) every point of the spectrum of S is the limit of a sequence of eigenvalues from the spectrum of the individual members of {S r } (ii) in the case when S is regular or limit-circle at each endpoint, a convergent sequence of eigenvalues from the individual members of {S r } has to converge to an eigenvalue of S (iii) in the general case when S is bounded below, property (ii) holds for all eigenvalues below the essential spectrum of S.