Transmutation operators and a new representation for solutions of perturbed Bessel equations

TL;DR

This paper introduces new integral kernel and solution representations for perturbed Bessel equations, enhancing numerical computation of eigenvalues and eigenfunctions especially for large parameters.

Contribution

It provides a novel Fourier-Jacobi series representation of the integral kernel and a Neumann series solution, improving accuracy for large nd nd non-integer nd nd enabling efficient numerical eigendata computation.

Findings

Fourier-Jacobi series for the integral kernel

Neumann series of Bessel functions for solutions

Enhanced numerical stability for large parameters

Abstract

New representations for an integral kernel of the transmutation operator and for a regular solution of the perturbed Bessel equation of the form $-u^{\prime\prime}+\left(\frac{\ell(\ell+1)}{x^{2}}+q(x)\right)u=\omega^{2}u$ are obtained. The integral kernel is represented as a Fourier-Jacobi series. The solution is represented as a Neumann series of Bessel functions uniformly convergent with respect to $\omega$. For the coefficients of the series convenient for numerical computation recurrent integration formulas are obtained. The new representation improves the ones from arXiv:1609.06679 and arXiv:1712.01363 for large values of $\omega$ and $\ell$ and for non-integer values of $\ell$. The results are based on application of several ideas from the classical transmutation (transformation) operator theory, asymptotic formulas for the solution, results connecting the decay rate of the Fourier transform with the smoothness of a function, the Paley-Wiener theorem and some results from constructive approximation theory. 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.