= Static Yukawa Green function
{title2=$G_\kappa(r)=e^{-\kappa r}/(4\pi r)$}
Let $\kappa>0$ be an inverse length. In three spatial dimensions, seek the <Green's function> that decays at infinity and satisfies
$$
(-\nabla^2+\kappa^2)G_\kappa(\mathbf r)=\delta^{(3)}(\mathbf r),\qquad r=|\mathbf r|,
$$
where $\nabla^2$ is the <Laplacian> and $\delta^{(3)}$ the three-dimensional <Dirac delta function>. The <point source> and the <differential operator> have <spherical symmetry>. For $r>0$, the <radial Laplacian> gives
$$
G_\kappa''+\frac2rG_\kappa'-\kappa^2G_\kappa=0.
$$
Writing $u(r)=rG_\kappa(r)$ reduces this to $u''-\kappa^2u=0$. Decay at infinity removes the growing <exponential function>, leaving $G_\kappa(r)=C e^{-\kappa r}/r$.
The <Dirac delta function> fixes the coefficient. Integrating over a ball $B_\epsilon$ and using the <divergence theorem> gives
$$
-4\pi\epsilon^2G_\kappa'(\epsilon)+\kappa^2\int_{B_\epsilon}G_\kappa\,d^3r=1.
$$
The first term tends to $4\pi C$, while the second is $O(\epsilon^2)$. Hence $C=1/(4\pi)$ and
$$
G_\kappa(r)=\frac{e^{-\kappa r}}{4\pi r}.
$$
This solves the source equation in the sense of <distributions>, including at the origin. Merely verifying the equation for $r>0$ would leave the normalization undetermined. The limit $\kappa\to0$ at fixed $r>0$ is $1/(4\pi r)$, the <Green function of the Laplacian> for the convention $-\nabla^2$.
For an attractive scalar toy interaction, absorb constants into $A>0$ and set
$$
V(r)=-A\frac{e^{-\kappa r}}r,\qquad F_r=-\frac{dV}{dr}=-A\frac{e^{-\kappa r}}{r^2}(1+\kappa r).
$$
Here $A$ has units of <energy> times length, and negative $F_r$ points inward. At equal $A$, comparison with the unscreened model $V_0=-A/r$ gives
$$
\frac{|V|}{|V_0|}=e^{-x},\qquad \frac{|F_r|}{|F_{0r}|}=(1+x)e^{-x},\qquad x=\kappa r.
$$
Thus the <potential energy> and <force> have the same exponential scale but different suppression factors. At $x=1$, these ratios are approximately $0.368$ and $0.736$; at $x=3$, they are $0.0498$ and $0.199$. Finite range means rapid decay, not an abrupt cutoff.
\Image[/yukawa-green-kernel-and-coulomb-decay.png]
{title=The Yukawa Green function and its decay relative to the Coulomb kernel}
{description=Original scalar-model calculation. Left: $4\pi G_\kappa/\kappa=e^{-x}/x$ compared with the unscreened $1/x$, for $x=\kappa r>0$. Right: relative <potential energy> and radial <force> magnitudes. The source singularity is excluded from the plotted interval; this is not a measured <nucleon> <potential energy> or a model of its core.}
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