  The Nekrasov-Shatashvili limit for the low-energy behavior of N=2 and N=2*
supersymmetric SU(2) gauge theories is encoded in the spectrum of the Mathieu
and Lam'e equations, respectively. This correspondence is usually expressed via
an all-orders Bohr-Sommerfeld relation, but this neglects non-perturbative
effects, the nature of which is very different in the electric, magnetic and
dyonic regions. In the gauge theory dyonic region the spectral expansions are
divergent, and indeed are not Borel-summable, so they are more properly
described by resurgent trans-series in which perturbative and non-perturbative
effects are deeply entwined. In the gauge theory electric region the spectral
expansions are convergent, but nevertheless there are non-perturbative effects
due to poles in the expansion coefficients, and which we associate with
worldline instantons. This provides a concrete analog of a phenomenon found
recently by Drukker, Marino and Putrov in the large N expansion of the ABJM
matrix model, in which non-perturbative effects are related to complex
space-time instantons. In this paper we study how these very different regimes
arise from an exact WKB analysis, and join smoothly through the magnetic
region. This approach also leads to a simple proof of a resurgence relation
found recently by Dunne and Unsal, showing that for these spectral systems all
non-perturbative effects are subtly encoded in perturbation theory, and
identifies this with the Picard-Fuchs equation for the quantized elliptic
curve.
