  We study various non-perturbative approaches to the quantization of the
Seiberg-Witten curve of ${\cal N}=2$, $SU(2)$ super Yang-Mills theory, which is
closely related to the modified Mathieu operator. The first approach is based
on the quantum WKB periods and their resurgent properties. We show that these
properties are encoded in the TBA equations of Gaiotto-Moore-Neitzke determined
by the BPS spectrum of the theory, and we relate the Borel-resummed quantum
periods to instanton calculus. In addition, we use the TS/ST correspondence to
obtain a closed formula for the Fredholm determinant of the modified Mathieu
operator. Finally, by using blowup equations, we explain the connection between
this operator and the $\tau$ function of Painleve $\rm III$.
