Anton Zettl

Computing Eigenvalues of Singular Sturm-Liouville Problems

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.

Algorithm 810: The SLEIGN2 Sturm-Liouville Code

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.

Regular approximations of singular Sturm-Liouville problems

Given any self-adjoint realization S of a singular Sturm-Liouville (S-L) problem, it is possible to construct a sequence {Sr{ 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 {Sr{(ii) in the case when 5 is regular or limit-circle at each endpoint, a convergent sequence of eigenvalues from the individual members of {Sr{ 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.

Integral Inequalities and Second Order Linear Oscillation

on the interval [0, a), u 0. Below, by a solution we mean a non-trivial solution. A solution is oscillatory if it has an infinite number of zeros. Numerous oscillation criteria are known and diversified approaches have proved fruitful. Among these is the use of the associated Riccati equation. Although the theory of differential and integral inequalities has been used by some investigators, see, e.g., Hartman [4, 71, the authors believe that it has not been fully exploited. The non-oscillation of (1.1) is equivalent to the existence of a solution of the associated Riccati equation on a half line. Thus one technique of proving oscillation is to show that any solution of the Riccati equation diverges to i-co at a finite point. The theory of differential and integral inequalities comes in handy in the estimation of the growth of such a solution. Below we describe some known results that are relevant to results established in this paper. Although these are stated here for the case (I = co, in the rest of the paper we consider both cases a < 0~) and a = 00. It is well known that if q is large in the mean then (1.1) is oscillatory. A typical example of such a criterion is due to Fite, Leighton and Wintner. If

Norm inequalities for derivatives and differences

Norm inequalities of product form relating a function and two of its derivatives and a sequence and two of its differences are discussed and compared. This paper contains a discussion of the possible values of the p and q norms of a function y and its n/sup th/ derivative y/sup (n)/. Detailed elementary proofs of the basic inequality for functions and a number of related inequalities are also given. A discussion of the growth at infinity of derivatives and a summary of cases when the best constant is known explicitly for both the continuous and the discrete versions of the basic inequality are finally discussed. 85 refs., 2 figs., 1 tab.

Adjoint and Self-Adjoint Boundary Value Problems with Interface Conditions

In this paper the class of adjoint and in particular self-adjoint boundary value problems associated with ordinary linear differential equations is extended to include problems which may have discontinuities in the solution or some of its derivatives at a finite number of interior points.

Eigenvalue and eigenfunction computations for Sturm-Liouville problems

SLEIGN is a software package for the computation of eigenvalues and eigenfunction:s of regular and singular Sturm Liouville boundary value problems, The package is a modification and extension of a code with the same name developed by Bailey, Gordon, and Shampinej which is described in ACM Z’OMS 4 (1978), 193-208. The modifications and extensions include (1) a restructuring of the FORTRAN program, (2) the coverage of problems with semidefi nite weight functions, and (3) the coverage of problems with indefinite weight functions.

Full- and partial-range eigenfunction expansions for Sturm-Liouville problems with indefinite weights *

This article is concerned with eigenvalue problems of the form Au = λTu in a Hilbert space H, where Ais a selfadjoint positive operator generated by a second-order Sturm-Liouville differential expression and T a selfadjoint indefinite multiplicative operator which is one-to-one. Emphasis is on the full-range and partial-range expansionproperties of the eigenfunctions.

Discontinuous dependence of the n-th Sturm-Liouville eigenvalue

The n-th eigenvalue of a regular Sturm-Liouville problem is not a continuous function of the boundary conditions. On the other hand if the index n is allowed to “jump”, then each eigenvalue can be embedded in an eigenvalue “branch” which is not only continuous but differentiable.

Ramifications of Landau's inequality

The problem of determining the best constant κ in the inequality ‖ y ′‖≦ K ‖ y ‖ ‖ y ″‖ is discussed in the context of the classical L p spaces, 1 ≦ p ≦ ∞.