Eigenvalue problems, spectral parameter power series, and modern applications

• Published in: Mathematical methods in the applied sciences Vol. 38; no. 10; pp. 1945 - 1969
• Main Authors: Khmelnytskaya, Kira V., Kravchenko, Vladislav V., Rosu, Haret C.
• Format: Journal Article
• Language: English
• Published: Blackwell Publishing Ltd 15.07.2015
• Subjects: Approximation; Eigenvalues; Mathematical analysis; Mathematical models; periodic coefficients; Power series; Representations; Schrödinger equation; Spectra; spectral parameter power series; spectral problem; Sturm-Liouville problem; Taylor series
• ISSN: 0170-4214; 1099-1476
• DOI: 10.1002/mma.3213

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 leads to a powerful numerical method, but most important, it offers a different insight into the spectral and transmission problems. Copyright © 2014 John Wiley & Sons, Ltd.