  Based on some recent progress on a relation between four dimensional super
Yang-Mills gauge theory and quantum integrable system, we study the asymptotic
spectrum of the quantum mechanical problems described by the Mathieu equation
and the Lam\'{e} equation. The large momentum asymptotic expansion of the
eigenvalue is related to the instanton partition function of supersymmetric
gauge theories which can be evaluated by a combinatorial method. The
electro-magnetic duality of gauge theory indicates that in the parameter space
there are three asymptotic expansions for the eigenvalue, we confirm this fact
by performing the WKB analysis in each asymptotic expansion region. The results
presented here give some new perspective on the Floquet theory about periodic
differential equation.
