  The physical observables of quantum theory can be described by perturbation
theory, which is often given by diverging power series. This divergence is
connected to the existence of non-perturbative phenomena, where resurgence
allows us to study this connection. Applying this idea to the WKB expansion,
the exact WKB analysis gives a clear connection to non-perturbative phenomena.
In this paper, we apply the exact WKB analysis to the Unruh effect and Hawking
radiation. The mechanism we found in this paper is similar to the Schwinger
effect of a constant electric field, where the background is static but the
Stokes phenomenon appears in the temporal part. Comparing this with a sonic
black hole, our calculations show a clear discrepancy between them. Then, we
briefly explain how quantum backreactions can be included in the exact WKB
formalism.
