I know that the Feynman-Kac formula gives a representation of the solutions of PDEs of the form $$ \partial_t u = Lu + Vu$$ for some differential operator $L$ and a bounded potential $V$.
From the few quantum mechanics I studied I know that the wave function of an electron orbiting a fixed proton solves something like $$Lu + Vu = 0$$ with $L \sim \Delta$ and $V \sim 1/|x|$.
Is there a Feller semigroup representation of such functions?
To me this problem seems more similar representing the solution of Laplace's equation $$\begin{cases} \Delta u = 0 \text{ in } D \\ u = f \text{ in } \partial D\end{cases}$$ using $u(x) = \mathbb{E}_x (f(X_\tau))$ where $\tau$ is the hitting time of $D^c$ of Broniwan motion $X_t$ starting from $x$. In the Hydrogen atom case though the domain is all of $\mathbb{R}^3$, so no boundary condition is given, and the operator is not purely differential.