Spectral parameter power series for perturbed Bessel equations

TL;DR

This paper introduces a spectral parameter power series (SPPS) approach for solving singular Bessel type Sturm-Liouville problems with complex coefficients, providing new theoretical insights and a highly effective numerical method.

Contribution

It develops an SPPS representation for solutions and characteristic functions of perturbed Bessel equations, enabling a novel numerical method applicable to complex spectra and coefficients.

Findings

The SPPS-based method accurately computes complex eigenvalues.

It outperforms existing solvers like SLEIGN2 and MATSLISE on test problems.

The method effectively handles equations with spectral parameters at a differential operator.

Abstract

A spectral parameter power series (SPPS) representation for regular solutions of singular Bessel type Sturm-Liouville equations with complex coefficients is obtained as well as an SPPS representation for the (entire) characteristic function of the corresponding spectral problem on a finite interval. It is proved that the set of zeros of the characteristic function coincides with the set of all eigenvalues of the Sturm-Liouville problem. Based on the SPPS representation a new mapping property of the transmutation operator for the considered perturbed Bessel operator is obtained, and a new numerical method for solving corresponding spectral problems is developed. The range of applicability of the method includes complex coefficients, complex spectrum and equations in which the spectral parameter stands at a first order linear differential operator. On a set of known test problems we show that the developed numerical method based on the SPPS representation is highly competitive in comparison to the best available solvers such as SLEIGN2, MATSLISE and some other codes and give an example of an exactly solvable test problem admitting complex eigenvalues to which the mentioned solvers are not applicable meanwhile the SPPS method delivers excellent numerical results.