A Neumann series of Bessel functions representation for solutions of perturbed Bessel equations
Provides a novel numerical method for solving perturbed Bessel equations, which are important in mathematical physics and engineering, with improved accuracy and efficiency over existing methods.
The paper presents a new representation for solutions of perturbed Bessel equations as a Neumann series of Bessel functions, enabling efficient and accurate computation of large sets of eigendata with non-deteriorating accuracy.
A new representation for a regular solution of the perturbed Bessel equation of the form $Lu=-u"+\left( \frac{l(l+1)}{x^2}+q(x)\right)u=ω^2u$ is obtained. The solution is represented as a Neumann series of Bessel functions uniformly convergent with respect to $ω$. 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.