I have issues or confusions to solve the vector Helmholtz equation in free space for a spherically symmetric source. Let the equation:
$$ \left( \widehat{\Delta} + k^2 \right) \boldsymbol{a} = f(r) \boldsymbol{e_r}$$
with $k$ positive wave vector, $\widehat{\Delta}$ vector Laplacian, $r$ spherical radius, $f(r)$ a function (with $\int_0^{\infty} r^n f(r) d r < + \infty$ for integers $n$, for instance $f \propto e^{- \alpha r}$ with $\alpha >0$ is a perfect candidate), $\boldsymbol{a}$ the unknown and $\boldsymbol{e_r}$ the unit radial vector.
How to solve it just for free space ? In far field approximation it is largely good enough.
What I tried
Because the source is radial, I assume the field $\boldsymbol{a}$ should be radial: $ \boldsymbol{a}= a(r) \boldsymbol{e_r}$. Then I have (I checked my mistake in del formula article):
$$ \left( \Delta_r - 2/r^2 + k^2 \right) a(r) = f(r) $$
Then I was thinking using the Green's function of Helmholtz equation but it is not a scalar Helmholtz equation... Thanks for reading.
Hint
This last equation can be rewrite in:
$$ \left( u^2 \partial_u^2 + 2 u \partial_u - 2 + u^2 \right) a(u) = u^2 f(u) $$
with $u = k r$, is an inhomogeneous spherical Bessel differential equation of order $1$ ($n = 1$ in the link). We can add boundary conditions (it is even better than free space for my problem): $$a(0) = 0$$ $$a(+ \infty) \rightarrow 0$$ With these B.C, it is maybe possible to try an eigenvalue decomposition on spherical Bessel function of first kind (they are nul on $0$ for order $> 0$) on an order to define. Trying this I found (using Sturm-Liouvile and eigenvalue decomposition of Green function):
$$ a(r) = 1/k^2 \sum_{n=2}^{+ \infty} \int_0^{+ \infty} u'^2 f(u')\frac{j_n(u') j_n(k r)}{N_n \left[2 - n(n+1) \right]} d u' $$
with $N_n = \int_0^{+ \infty} j_n(u')^2 d u'$, $j_n(u)$ being the spherical Bessel function of first kind and order $n$. The issue is that the sum is incomplete (I miss the term $0$ and $1$), so I do not know if the Green function decomposition is still valid.