Infinitesimal Generators and Expectation of First Hitting Time as Solution of Differential Equation

I’ve been learning about Linear Diffusions and how their infinitesimal generators can be used to relate expectations and deterministic differential equations.

Let $X$ be an one-dimensional diffusion with the following dynamics:

$dX_t = mu(X_t)dt + sigma(X_t)dW_t$

Its infinitesimal generator $mathcal{G}$ takes the form of the following second-order differential operator:

$mathcal{G}f(x) = frac{1}{2}sigma^2(x)frac{d^2f}{dx^2}(x) + mu(x)frac{df}{dx}$

I’ve seen, for example, that when one defines: $u(x) = Pr( tau_a < tau_b | X_0 = x)$, with $a<b$ and $tau_a$ and $tau_b$ being their respective hitting times. With some regularity hypothesis under $u$ and by proceeding analogously to a "first step analysis" for discrete Markov Chains, one can show that $u$ is a solution of: $mathcal{G}u(x) = 0$, with $u(a) = 1$ and $u(b) = 0$ as boundary conditions.

I know this approach is much more general than this simple example and can be used also in the opposite direction. I think it is actually what is done for the Dirichlet’s Problem and the Feynman-Kac Formula.

In the article http://users.iems.northwestern.edu/~linetsky/cev.pdf, the authors claim that:

For such a $S$ having the following local volatility dynamics:

$dS_t = mu S_tdt + sigma(S_t)S_tdW_t$

The following is true:

$mathbb{E}[e^{-rtau_a}mathbb{1}_{{tau_a < infty}} | S_0 = x] = frac{phi_r(x)}{phi_r(a)}$, for $ x geq a$

and

$mathbb{E}[e^{-rtau_b}mathbb{1}_{{tau_b < tau_0 }} | S_0 = x] = frac{psi_r(x)}{psi_r(b)}$, for $ x leq b$

where: $phi_r$ and $psi_r$ are solutions of:
$mathcal{G}v = rv$.

$psi_r$ is required to be an increasing function and $phi_r$ is required to be a decreasing function such that: $psi(0+) = (0)$, if $0$ is a regular boundary point (we force 0 to be a killing boundary).

I have no idea how those expectations can be expressed in terms of the solutions of this "eigenvector equation". Could someone please explain to me why? Please, I ask you to be as didactic as possible, once I’m kinda new to the subject.