A Neumann series of Bessel functions representation for solutions of perturbed Bessel equations

TL;DR

This paper introduces a novel Neumann series of Bessel functions representation for solutions of perturbed Bessel equations, enabling efficient numerical computation of eigenvalues and eigenfunctions with high accuracy.

Contribution

It develops a new analytical solution representation using Neumann series of Bessel functions, with explicit formulas for coefficients based on transmutation operator theory and spectral parameter power series.

Findings

Uniform convergence of the series with respect to frequency parameter

Explicit formulas for series coefficients suitable for numerical implementation

Effective numerical method for computing eigenvalues and eigenfunctions

Abstract

A new representation for a regular solution of the perturbed Bessel equation of the form $Lu=-u"+\left( \frac{l(l+1)}{x^2}+q(x)\right)u=\omega^2u$ is obtained. The solution is represented as a Neumann series of Bessel functions uniformly convergent with respect to $\omega$. For the coefficients of the series explicit direct formulas are obtained in terms of the systems of recursive integrals arising in the spectral parameter power series (SPPS) method, as well as convenient for numerical computation recurrent integration formulas. The result is based on application of several ideas from the classical transmutation (transformation) operator theory, recently discovered mapping properties of the transmutation operators involved and a Fourier-Legendre series expansion of the transmutation kernel. For convergence rate estimates, asymptotic formulas, a Paley-Wiener theorem and some results from constructive approximation theory were used. We show that the analytical representation obtained among other possible applications offers a simple and efficient numerical method able to compute large sets of eigendata with a nondeteriorating accuracy.