Inverse Functions of the Products of Two Bessel Functions and Applications to Potential Scattering

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Abstract
Inverse functions of products of two Bessel functions jl(xy)jm(xy) are determined for the cases m = l, l + 1, and l + 2. Integral representations for these inverse functions in terms of Neumann functions are given, and some of the simplest ones are expressed in terms of trigonometric functions. We show how one may obtain an integral representation for any well‐behaved function in terms of products of two Bessel functions, with the help of these inverse functions and also outline some of their applications to potential scattering. In particular, we demonstrate the usefulness of the inverse functions in determining the potential explicitly from the phase shifts in the Born approximation.

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