  We study the problem of how the Floquet property manifests for periodic
Schr\"{o}dinger operators which are known to have multiple of asymptotic
spectral solutions. The main conclusions are made for elliptic potentials, we
demonstrate that for each period of the elliptic function there is a relation
about the Floquet exponent and the monodromy of wave function. Among them there
are two relations not explained by the classical Floquet theory. These
relations produce both old and new asymptotic solutions consistent with results
already known.
