Matching numbers are the basics Ito and McKean use to build out a bunch of stuff, like singular points and shunts. The four maching numbers $e_1, e_2, e_3, e_4$ are defined as
$e_1 = \lim_{b \downarrow a} E_a(e^{-m_b}) \\ e_2 = \lim_{b \downarrow a}\lim_{c \downarrow a} E_c(e^{-m_b}) \\ e_3 = \lim_{b \downarrow a} E_b(e^{-m_a}) \\ e_4 = \lim_{b \downarrow a} E_b(e^{-m_{a+}}) \\ $
where the diffusion process $x(t)$ starts at $a$ leading to $b$ and $c$, $a \leq b \leq c$. As usual $m_b$ is the first passage time to $b$, and $m_{a+} = \inf(t: x(t) > a)$.
My first question is why are these objects called "matching numbers" ? What are they being "matched"? and what's the intuition behind these construction?
I can see that $e_1$ and $e_3$ are kind of inverse of each other (but not quite since $x$ starts at $a$).
