This subject, especially 1D case, is based on Floquet Theory. It's important to study Mathieu operator's spectral properties; e.g asymptotics of its spectral gaps subject to some specific boundary conditions.
These spectral gaps is related with the notion of tunneling in momentum space. "Of particular interest are the gaps, which are the size of various forbidden regions of the spectrum in the one electron theory of solids or alternatively regions of instability in the theory of parametric resonance in classical mechanics." J. Avron- B. Simon.
The books of S. Winkler and W. Magnus, "The coexistence problem for Hill's equation"
and M. S. P. Eastham, "The spectral theory of periodic dierential operators" are the basics
for Floquet Theory and in particular, for the Mathieu operator.