Permissible Electrostatic Potential

Let us consider a $$1D$$ real function $$V(x)$$. When is this a classical electrostatic potential?

My take on the problem:

1. $$V(x)$$ must be differentiable everywhere. In fact, we should be able to differentiate it $$n$$ times.
2. $$V(x)$$ should vanish at $$\pm \infty$$.

I think these are necessary and sufficient conditions. Is this right? How do I deal with discrete charge distributions, where the potential is not differentiable at the points where the charges are present?

• Can you strictly define what is a classical electrostatic potential? Oct 15 '20 at 7:55
• @AndreasMastronikolis a physical potential that obeys poisson's equation. Oct 15 '20 at 9:36

I think that most conditions can be seen from the Poisson equation: $$\frac{d^2 V(x)}{dx^2} = -\rho(x).$$ Thus, it should be differentiable everywhere except a few singular points. One often bypasses this issue by using generalized functions (delta-function and Heaviside step function).
However, there is no requirement that the potential vanishes in infinity. For example, the potential corresponding to a constant electric field of magnitude $$E$$ is $$V(x) = - Ex$$
• Aren't you missing a negative sign in Poisson's Equation? (oh, and an $\epsilon_0$...) Oct 15 '20 at 7:50