Eigenvalue problems, spectral parameter power series, and modern applications

Abstract

In the present review we deal with the recently introduced method of spectral parameter power series (SPPS) and show how its application leads to an explicit form of the characteristic equation for different eigenvalue problems involving Sturm-Liouville equations with variable coefficients. We consider Sturm-Liouville problems on finite intervals; problems with periodic potentials involving the construction of Hill's discriminant and Floquet-Bloch solutions; quantum-mechanical spectral and transmission problems as well as the eigenvalue problems for the Zakharov-Shabat system. In all these cases we obtain a characteristic equation of the problem which in fact reduces to finding zeros of an analytic function given by its Taylor series. We illustrate the application of the method with several numerical examples which show that at present the SPPS method is the easiest in the implementation, the most accurate and efficient. We emphasize that the SPPS method is not a purely numerical technique. It gives an analytical representation both for the solution and for the characteristic equation of the problem. This representation can be approximated by different numerical techniques and for practical purposes constitutes a powerful numerical method but most important it offers additional insight into the spectral and transmission problems.

44 pages, 6 figures, 102 references, revised and some updated references

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• On the Singular Weyl-Titchmarsh Function of Perturbed Spherical Schroedinger Operators
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Cited by in corpus (6)

• Analytic approximation of transmutation operators and applications to highly accurate solution of spectral problems
• Spectral parameter power series for perturbed Bessel equations
• Construction of transmutation operators and hyperbolic pseudoanalytic functions
• Transmutations and spectral parameter power series in eigenvalue problems
• Modified spectral parameter power series representations for solutions of Sturm-Liouville equations and their applications
• Spectral problems in inhomogeneous media, spectral parameter power series and transmutation operators