Units of Dirac delta potential constant

Question

Suppose we have a Dirac delta potential correction of the form $$ V(x)=V_0\delta(x-x_0)$$ What would be the units of $V_0$ ? I think it should be units of energy as calculating first order perturbation for the energy of an eigenstate is $$\Delta E=\langle\psi|V_0\delta(x-x_0)|\psi\rangle=V_0|\psi(x_0)|^2$$ Evidently implying $[V_0]$=Joules

Answer 1

$\int \delta(x) dx=1$ shows that $\delta(x)$ has units of (length)$^{-1}$. As $V\delta(x)$ is a potential energy, $V$ must have units of (energy)$\times$ (length). This is consistent with the first-order energy shift as $|\psi|^2$ is probability (dimensionless) per unit length.