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.

Each finite p-perfect group G (p a prime) has a universal central p-extension coming from the p part of its Schur multiplier. Serre gave a Stiefel-Whitney class approach to analyzing spin covers of alternating groups (p = 2) aimed at geometric covering space problems that included their regular realization for the Inverse Galois Problem. A special case of a general result is that any finite simple group with a nontrivial p part to its Schur multiplier has an infinite string of perfect centerless group covers exhibiting nontrivial Schur multipliers for the prime p. Sequences of moduli spaces of curves attached to G and p, called Modular Towers, capture the geometry of these many appearances of Schur multipliers in degeneration phenomena of Harbater-Mumford cover representatives. These are modular curve tower generalizations. So, they inspire conjectures akin to Serre's open image theorem, including that at suitably high levels we expect no rational points. Guided by two papers of Serre's, these cases reveal common appearance of spin structures producing θ-nulls on these moduli spaces. The results immediately apply to all the expected Inverse Galois topics. This includes systematic exposure of moduli spaces having points where the field of moduli is a field of definition and other points where it is not.

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 describe a new algorithm to compute the eigenvalues of singular Sturm-Liouville problems with separated self-adjoint boundary conditions for both the limit-circle nonoscillatory and oscillatory cases. Also described is a numerical code implementing this algorithm and how it compares with SLEIGN. The latter is the only effective general purpose software available for the computation of the eigenvalues of singular Sturm-Liouville problems.

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