For a generic one-dimensional potential, the WKB approximation yields the quantization condition
$$ \oint p dq = (n + 1/2)\hbar . $$
Here, the correction factor $1/2 $ was obtained by Kramers by studying the behavior of the wave function near the turning point. It is a study in the realm of differential equation.
However, it seems that Maslov got it in quite a different way. It looks like that his approach is more topological.
So, could anyone explain Maslov's method just in the 1d setting? I know he has a book on it. But that is far beyond my comprehension.