MATHEMATICS OF COMPUTATION Volume 68, Number 227, Pages 1057–1066 S 0025-5718(99)01053-4 Article electronically published on February 10, 1999 EIGENVALUES OF PERIODIC STURM-LIOUVILLE PROBLEMS BY THE SHANNON-WHITTAKER SAMPLING THEOREM AMIN BOUMENIR Abstract. We are concerned with the computation of eigenvalues of a peri- odic Sturm-Liouville problem using interpolation techniques in Paley-Wiener spaces. We shall approximate the Hill discriminant by sampling a few of its values and then find its zeroes which are the square roots of the eigenvalues. Computable error estimates are provided together with eigenvalue enclosures. 1. Introduction We would like to introduce a new method for computing the eigenvalues of classical regular Sturm-Liouville problems with periodic boundary conditions  −ϕ ′′ (t,μ)+ Q(t)ϕ(t,μ)= μ 2 ϕ(t,μ) t ∈ [0,ω] ϕ(0,μ)= ϕ(ω,μ) and ϕ ′ (0,μ)= ϕ ′ (ω,μ), (1.1) where Q(t) ∈ L 1 (0,ω). For the spectral theory of periodic differential equations, we shall refer to [4], [1] and [8]. Recall that in general the spectrum of (1.1) may not be simple as in the case of separated boundary conditions, and this is a major difficulty. In this work, we shall extend the method in [3], which relies on the interpola- tion in Paley-Wiener spaces of a certain boundary function, whose zeros are square roots of eigenvalues. In our case the boundary function turns out to be the well- known Hill discriminant (see [4]) and only few values are needed for its approxima- tion. The truncation error can be minimized by increasing the number of sampling points, which gives higher accuracy on the numerical side and provides eigenvalue enclosures. We shall examine a few examples, where eigenvalue enclosures and com- parisons of the results with the well-known code Sleign2 are provided. In the last example our interpolation scheme is compared with an “exact” solution. One of the features of the method is the possibility of locating double eigenvalues. Indeed, by minimizing the truncation error, we can zoom in on close eigenvalues, and if they happen to be simple, then we can find two disjoint enclosures separating them. 2. Hill’s discriminant The Hill discriminant, which is at the heart of the spectral theory of the periodic Sturm-Liouville operator, is defined by Δ(μ) := ϕ 1 (ω,μ)+ ϕ ′ 2 (ω,μ), Received by the editor June 9, 1997 and, in revised form, October 28, 1997. 1991 Mathematics Subject Classification. Primary 34L15, 42A15. Key words and phrases. Periodic Sturm-Liouville problems, eigenvalue problem, interpolation, Shannon-Whittaker sampling theorem. c 1999 American Mathematical Society 1057 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use