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.

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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.

We develop the left-definite analysis associated with the self-adjoint Jacobi operator A ( , ) k , generated from the classical secondorder Jacobi differential expressionin the Hilbert space L 2 , (−1, 1) := L 2 ((−1, 1); w , (t)), where w , (t) = (1 − t) (1 + t) , that has the Jacobi polynomials {P ( , ) m } ∞ m=0 as eigenfunctions; here, , > − 1 and k is a fixed, non-negative constant. More specifically, for each n ∈ N, we explicitly determine the unique left-definite Hilbert-Sobolev space W ( , ) n,k (−1, 1) and the corresponding unique left-definite selfadjoint operator B ( , )

SynopsisThis paper considers properties of the spectrum of differential operators derived from differential expressions of the second order. The object is to link the spectral properties of these differential operators with the analytic, function-theoretic properties of the solutions of the differential equation. This provides an alternative approach to the spectral theory of these differential operators but one which is consistent with the standard definitions used in Hilbert space theory. In this way the approach may be of interest to applied mathematicians and theoretical physicists.