General relationship between Green's functions and propagators?

Question

Given a linear operator $\hat{L}$ and $x \in \mathbb{R}^n$, a Green's function of $\hat{L}$ is a function that satisfies $$\hat{L}_xG(x,x^\prime) = \delta^{(n)}(x-x^\prime)$$ A "propagator" $G(x,x^\prime,t,t^\prime)$ of a linear PDE is essentially the integral kernel of the evolution operator. That is, we assign one variable to be the "time" variable $t$ and then if $\psi(x,t)$ solves the PDE then for $t>t^\prime$ $$\psi(x,t) = \int_{\mathbb{R}^{n-1}} G(x,x^\prime,t,t^\prime) \psi(x^\prime,t^\prime) d^{n-1}x^\prime$$ (where the dependence on $t^\prime$ on the RHS drops out). The Green's function and the propagator seem like very different objects but it turns out that if our PDE is $$\hat{L}\psi(x,t) = 0$$ where $$\hat{L} = i \frac{\partial}{\partial t} - \hat{H}_x$$ the two objects are the same. This is a standard result in quantum mechanics. This seems like a big "coincidence" and the proof relies heavily on the above form that $\hat{L}$ takes. Is there is a general relationship between these two objects? Could there be situations in which the two are different?

Answer 1

The answer to my question is Duhamel's principle.