fa.functional analysis – Poisson’s Equation, the Dirac Distribution, and the Hessian Matrix Answer

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fa.functional analysis – Poisson’s Equation, the Dirac Distribution, and the Hessian Matrix

The Poisson equation is
$$\nabla^2 \Phi(\bar{r}) = 4\pi\rho(\bar{r})$$

The simplest potential is that of a point mass, for which the potential is of the form $$-1/|\bar{r}|$$ and the density function is the Dirac delta distribution $$\delta(\bar{r})$$. Now, my question is as follows: if I were to instead take the Hessian of that potential (eg in Cartesian coordinates), I should be able to take the trace and recover the Laplacian of the potential. So, I should find that the trace of the Hessian matrix gives me a delta distribution, but instead I find that the trace equals 0. I understand that this discrepancy arises from the fact that the Dirac distribution doesn’t behave like a normal function, but I’m curious how I would be able to resolve these seemingly inconsistent results. I’ve also tried computing the Hessian for this function in a basis-independent way using matrix calculus, but the results turn out to be the same.

Since $$1/r$$ is the fundamental solution of the Poisson equation, this is giving me quite a lot of trouble when I try to generalize computing the Hessian to more complicated potentials.

Sorry if this question isn’t well-phrased. My background is not in mathematics.

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