Numerical computation of eigenvalues in spectral gaps of Sturm-Liouville operators
The numerical calculation of the eigenvalues of a singular Sturm-Liouville problem \(-y'' + Q(x)y = \lambda y, x \in R+\), is studied in the case that the potential \(Q\) is a decaying \(L_1\) perturbation of a periodic function. Hence its essential spectrum has a band-gap structure. Two approaches are considered. One uses a shooting method, the other one an algebraic technique which is based on a theorem by \textit{G. Stolz} and \textit{J. Weidmann} [J. Reine Angew. Math. 445, 31--44 (1993; Zbl 0781.34052)]. Both approaches are capable of generating approximations to eigenvalues inside any gap of the essential spectrum. They do not generate spurious eigenvalues. Two examples with numerical ressults are given.
scientific article; zbMATH DE number 5015145