Numerical computation of eigenvalues in spectral gaps of Sturm–Liouville operators
Copyright policy )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.
- University of Eastern Piedmont Amadeo Avogadro Italy
- Cardiff University United Kingdom
- University of Pisa Italy
Numerical solution of boundary value problems involving ordinary differential equations, numerical examples, essential spectrum, Sturm-Liouville operator, eigenvalue problem, Applied Mathematics, shooting method, 510, Sturm-Liouville theory, Computational Mathematics, Sturm-Liouville operator, Eigenvalue problem, spectral gap, eigenvalue problem, Sturm–Liouville operator, Numerical approximation of eigenvalues and of other parts of the spectrum of ordinary differential operators, Numerical solution of eigenvalue problems involving ordinary differential equations
Numerical solution of boundary value problems involving ordinary differential equations, numerical examples, essential spectrum, Sturm-Liouville operator, eigenvalue problem, Applied Mathematics, shooting method, 510, Sturm-Liouville theory, Computational Mathematics, Sturm-Liouville operator, Eigenvalue problem, spectral gap, eigenvalue problem, Sturm–Liouville operator, Numerical approximation of eigenvalues and of other parts of the spectrum of ordinary differential operators, Numerical solution of eigenvalue problems involving ordinary differential equations
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