Analytic expressions for integrals of products of spherical Bessel functions

Abstract

Integrals of several spherical Bessel functions occur frequently in nuclear physics. They are difficult to evaluate using standard numerical techniques, because of their slowly decreasing oscillatory form. The authors derive an analytic expression for the infinite integral of three spherical Bessel functions. They then use this result, together with the closure relation for spherical Bessel functions, to show how in principle one can derive an analytic expression for the integral of any number of spherical Bessel functions. They demonstrate this by deriving an analytic expression for the integral of four spherical Bessel functions. As with all of these analytic formulae, the results require that all angular momenta corresponding to the spherical Bessel functions can be coupled together to give an overall scalar quantity and conserve parity. The authors discuss the numerical accuracy and stability of this procedure.