The Integral Over 2 Spherical Bessel Functions Multiplied by a Gaussian

TL;DR

This paper derives an analytical expression for an integral involving two spherical Bessel functions multiplied by a Gaussian, which is useful in nuclear physics calculations involving harmonic oscillator wavefunctions and Gaussian potentials.

Contribution

The paper provides a new analytical solution for a complex integral involving spherical Bessel functions and a Gaussian, aiding nuclear scattering computations.

Findings

Integral expressed as a finite sum over modified spherical Bessel functions.

Facilitates calculations in nuclear scattering with harmonic oscillator wavefunctions.

Enables efficient evaluation of momentum space matrix elements for Gaussian potentials.

Abstract

In this paper, the integral $\pmatrix{\lambda_1 &\lambda_2 &\lambda_3\cr 0 &0 &0\cr}\, \int_0^\infty \, r^{\lambda_3+2}\, \exp{(-\alpha r^2)}\, j_{\lambda_1}(k_1r) \,j_{\lambda_2}(k_2r) \,dr$, 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.