  We study periodic magnetic Schr\"odinger operators on covers of closed
manifolds in relation to Ma\~n\'e's critical energy values of the corresponding
classical Hamiltonian systems. In particular, we show that if the covering
transformation group is amenable, then the bottom of the spectrum is bounded
from above by Ma\~n\'e's critical energy value. We also determine the spectra
for various homogeneous spaces with left-invariant magnetic fields.
