Laplacian of $1/r^2$ (context: electromagnetism and Poisson equation)

I) Problem. We put (four pi times) the permittivity $4\pi\varepsilon=1$ equal to one (in the SI unit system) from now on for simplicity. Let us first rephrase OP's question a bit. Instead of starting from the potential

$$ \Phi~=~\frac{1}{r} \qquad \mathrm{and} \qquad \Phi~=~\frac{1}{2r^2}, \qquad r\neq 0,\tag{1} $$

respectively, let us assume that the electric field has be given as

$$ \vec{E}~=~\frac{\vec{r}}{r^3} \qquad \mathrm{and} \qquad\vec{E}~=~ \frac{\vec{r}}{r^4},\qquad r\neq 0,\tag{2}$$

respectively, and we want to know the charge density

$$ \rho~=~\vec{\nabla} \cdot \vec{E},\tag{3}$$

in particular, at the origin $r=0$. For $r\neq 0$, the charge density (3) is

$$ \rho~=~0 \qquad \mathrm{and} \qquad \rho~=~-\frac{1}{r^4},\qquad r\neq 0,\tag{4}$$

respectively. This rephrasing is just so we only have to differentiate one time instead of two times, but the argument is in principle the same, cf. footnote $1$.

II) Distribution theory and integration theory. An issue is that the electric field $\vec{E}$ has not been specified at $r=0$ in eq. (2). A way to make sense of this is to use distributions and test functions $f\in C^{\infty}_c(\mathbb{R}^3)$, i.e., infinitely often differentiable functions $f$ with compact support. We now declare that a smeared electric field $E^i$ is

$$ E^i[f]~:=~\int_{\mathbb{R}^3} \!d^3r \ E^i(\vec{r})f(\vec{r}).\tag{5}$$

A Lebesgue majorant of the integrand (5) is

$$ \frac{|f(\vec{r})|}{r^2} \qquad \mathrm{and} \qquad \frac{|f(\vec{r})|}{r^3},\qquad r\neq 0,\tag{6}$$

respectively. Only the first case of (6) is Lebesgue integrable in $\mathbb{R}^3$. This is the heart of the problem. In the second case, even after smearing with a test function $f$, the electric field $E^i$ does not make sense as a distribution.

III) In distribution theory, the derivative of a distribution is always defined by applying the derivative to the test function with a minus sign$^{1}$. If $E^i$ is a distribution (5), we can carry out the next step to define the derivative of $E^i$,

$$-\vec{\nabla} \cdot \vec{E}[f] ~:=~ \int_{\mathbb{R}^3} \!d^3r \ \vec{E}(\vec{r})\cdot\vec{\nabla} f(\vec{r}) ~\stackrel{\begin{matrix}\text{Leb. dom.}\\ \text{conv. thm.}\end{matrix}}{=}~ \lim_{\varepsilon\to 0^+} \int_{\{r\geq\varepsilon\}} \!d^3r \ \vec{E}(\vec{r})\cdot\vec{\nabla} f(\vec{r}) $$ $$ ~\stackrel{\begin{matrix}\text{int. by}\\ \text{parts}\end{matrix}}{=}~ \lim_{\varepsilon\to 0^+} \int_{\{r\geq\varepsilon\}} \!d^3r \left[ \vec{\nabla} \cdot\left( \vec{E}(\vec{r})f(\vec{r})\right)-f(\vec{r}) \underbrace{\vec{\nabla} \cdot \vec{E}(\vec{r})}_{=0}\right] $$ $$ ~\stackrel{\begin{matrix}\text{div.}\\ \text{thm.}\end{matrix}}{=}~ -\lim_{\varepsilon\to 0^+} \int_{\{r=\varepsilon\}} \!d^2\vec{A} \cdot \vec{E}(\vec{r})f(\vec{r}) ~\stackrel{(2)}{=}~-\lim_{\varepsilon\to 0^+} \int_{\{r=\varepsilon\}} \!\frac{d^2A}{\varepsilon^2}f(\vec{r}) $$ $$~=~-4\pi f(0),\tag{7} $$

where we performed the well-known manipulations in the first case to show that

$$ \vec{\nabla} \cdot\frac{\vec{r}}{r^3} ~=~ 4\pi\delta^3(\vec{r}).\tag{8} $$

IV) Regularization. Well, so much for distribution theory and mathematical idealization in Sections II-III. In reality, the $\frac{-1}{r^4}$ charge density, $r\neq 0$, in eq. (4) would break down as we approach the singularity $r\to 0$, so that we never get to ask: What sits at $r=0$? This leads to the idea of regularization

$$ \Phi_{\varepsilon}~=~\frac{1}{\sqrt{r^2+\varepsilon}} \qquad \mathrm{and} \qquad \Phi_{\varepsilon}~=~\frac{1}{2(r^2+\varepsilon)}, \qquad \varepsilon>0,\tag{9}$$

respectively, of eq. (1). The regularized charge density

$$ \rho_{\varepsilon}~=~-\vec{\nabla}^2\Phi_{\varepsilon}~\in~ C^{\infty}(\mathbb{R}^3),\qquad \varepsilon>0,\tag{10}$$

is

$$ \rho_{\varepsilon}~=~\frac{3\varepsilon}{(r^2+\varepsilon)^{\frac{5}{2}}} \qquad \mathrm{and} \qquad \rho_{\varepsilon}~=~\frac{3\varepsilon-r^2}{(r^2+\varepsilon)^{3}} ~=~\frac{4\varepsilon}{(r^2+\varepsilon)^{3}}-\frac{1}{(r^2+\varepsilon)^{2}},\tag{11}$$

respectively. We can now smear with a test function $f$. One may check that the first charge density $\rho_{\varepsilon}$ in eq. (11) satisfies

$$\lim_{\varepsilon\to 0^+}\rho_{\varepsilon} ~=~4\pi\delta^3(\vec{r}) , \tag{12} $$

while the second charge density $\rho_{\varepsilon}$ in eq. (11) does not make sense as a distribution when $\varepsilon\to 0^+$, i.e. after smearing with a test function $f$, the limit $\varepsilon\to 0^+$ is not finite.

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$^{1}$ In the second case, we could in principle define the smeared potential

$$ \Phi[f] ~:=~ \int_{\mathbb{R}^3} \!d^3r \ \Phi(\vec{r})f(\vec{r}),\tag{13}$$

because $\Phi=\frac{1}{2r^2}$, $r\neq 0$, is locally Lebesgue integrable in $\mathbb{R}^3$, and then define the electric field as a distribution

$$ \vec{E}[f] ~:=~ \Phi[\vec{\nabla} f] ~\stackrel{(13)}{=}~ \int_{\mathbb{R}^3} \!d^3r \ \Phi(\vec{r})\vec{\nabla}f(\vec{r}),\tag{14}$$

so that

$$ -\vec{\nabla} \cdot \vec{E}[f]~=~ \int_{\mathbb{R}^3} \!d^3r \ \Phi(\vec{r})\vec{\nabla}^2 f(\vec{r})~=~ \lim_{\varepsilon\to 0} \int_{\{r\geq\varepsilon\}} \!d^3r \ \Phi(\vec{r})\vec{\nabla}^2 f(\vec{r}).\tag{15}$$

However, this mathematical construct is not as useful in practice as one might naively have hoped for. For instance, if we try to integrate by parts, we essentially get back to the problem that the electric field $\vec{E}= \frac{\vec{r}}{r^4}$, $r\neq 0$, from eq. (2) is not locally Lebesgue integrable in $\mathbb{R}^3$.

The electric field from your potential is:

$$E(r) = {2\over r^3}$$

Using Gauss's law, the total charge in a sphere of radius R is:

$$Q(r) = \oint E \cdot dS = 4\pi r^2 {2\over r^3} = {8\pi\over r}$$

The total charge is decreasing with r, so there is a negative charge cloud of density

$$ \rho(r) = {1\over 4\pi r^2} {dQ\over dr} = - {4\over r^4}$$

But the total charge at infinity is zero, so there is a positive charge at the origin, cancelling the negative charge cloud, of a divergent magnitude. If you assume this charge is a sphere of infinitesimal radius $\epsilon$, the positive charge at the origin is

$$Q_0 = \int_\epsilon^\infty 4\pi r^2 {4\over r^4} = {16\pi \over \epsilon}$$

This is not a distribution in the mathematical sense, but it is certainly ok to work with, so long as you keep the $\epsilon$ around and take the limit $\epsilon$ goes to zero at the end of the day. Mathematicians have not had the last word on the class of appropriate generalized solutions yet.

Vladimir's answer is off by factor of 2. The laplacian is $\nabla^2(\frac{1}{r^2}) = \frac{4}{r^4}$ A potential that falls of as $\frac{1}{r^2}$ is a dipole (In general, if it falls off as $r^{-n}$ its an ($2^{n-1}$)-pole, e.g $\frac{1}{r^3}$ bheaviour is quadrupole, etc).

Is this a dirac delta? To find out, check : $$\int_{\mbox{All space}}\nabla^2(\frac{1}{r^2})d^3r $$ $$=16\pi\int_0^\infty \frac{1}{r^2}dr\neq 1$$ Yup, integral diverges, so it is NOT a delta function.

I think your confusion is regarding the nature of a delta function. If something blows up at the origin it does not mean it is necessarily a delta function.