The Integral Over 2 Spherical Bessel Functions Multiplied by a Gaussian

Abstract

In this paper, the integral $\pmatrix{λ_1 &λ_2 &λ_3\cr 0 &0 &0\cr}\, \int_0^\infty \, r^{λ_3+2}\, \exp{(-αr^2)}\, j_{λ_1}(k_1r) \,j_{λ_2}(k_2r) \,dr$, where $k_1$, $k_2$ and $\alpha$ are positive, is evaluated analytically. The result is a finite sum over the modified spherical Bessel function of the first kind. This result will be useful for nuclear scattering calculations, where harmonic oscillator nuclear wavefunctions are used or when evaluating momentum space matrix elements for a Gaussian potential.

7 pages