Neumann series of Bessel functions for the solutions of the Sturm-Liouville equation in impedance form and related boundary value problems

We present a Neumann series of spherical Bessel functions representation for solutions of the Sturm–Liouville equation in impedance form $$(\kappa (x)u^{\mathrm{\prime }}{)}^{\mathrm{\prime }}+\lambda \kappa (x)u=0,0<x<L,$$ in the case where $\kappa \in W^{1,2}(0,L)$ and has no zeros on the interval of interest. The $x$-dependent coefficients of this representation can be constructed explicitly by means of a simple recursive integration procedure. Moreover, we derive bounds for the truncation error, which are uniform whenever the spectral parameter $\rho =\sqrt{\lambda }$ satisfies a condition of the form $\mathrm{\mid }\mathrm{Im}\rho \mathrm{\mid }\le C$. Based on these representations, we develop a numerical method for solving spectral problems that enables the computation of eigenvalues with non-deteriorating accuracy.