Spectral parameter power series for Sturm-Liouville equations with a potential polynomially dependent on the spectral parameter and Zakharov-Shabat systems

Abstract

Abstract. A spectral parameter power series (SPPS) representation for solutions of Sturm-Liouville equations of the formis obtained. It allows one to write a general solution of the equation as a power series in terms of the spectral parameter λ. The coefficients of the series are given in terms of recursive integrals involving a particular solution of the equation (pu 0 ) +qu0 = 0. The convenient form of the solution of ( * ) provides an efficient numerical method for solving corresponding initial value, boundary value and spectral problems. A special case of the Sturm-Liouville equation ( * ) arises in relation with the Zakharov-Shabat system. We derive an SPPS representation for its general solution and consider other applications as the one-dimensional Dirac system and the equation describing a damped string. Several numerical examples illustrate the efficiency and the accuracy of the numerical method based on the SPPS representations which besides its natural advantages like the simplicity in implementation and accuracy is applicable to the problems admitting complex coefficients, spectral parameter dependent boundary conditions and complex spectrum.

Effective numerical method of spectral analysis of quantum graphs

“…Nonetheless, it is always possible to calculate increasingly higher approximate eigenvalues for a fixed M by shifting the spectral parameter [37]. The numerical performance of this technique is proven in the works [13,31,38].…”