  We study Hill's differential equation with potential expressed by elliptic
functions which arises in some problems of physics and mathematics. Analytical
method can be applied to study the local properties of the potential in
asymptotic regions of the parameter space. The locations of the saddle points
of the potential are determined, the locations of turning points can be
determined too when they are close to a saddle point. Combined with the
quadratic differential associated with the differential equation, these local
data give a qualitative explanation for the asymptotic eigensolutions obtained
recently. A relevant topic is about the generalisation of Floquet theorem for
ODE with doubly-periodic elliptic function coefficient which bears some new
features compared to the case of ODE with real valued singly-periodic
coefficient. Beyond the local asymptotic regions, global properties of the
elliptic potential are studied using numerical method.
