Copyright © 2010 Meirong Zhang. This is an open access article distributed under the
Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Given a periodic, integrable potential , we will study conditions on so that
the operator admits the maximum principle or the antimaximum principle
with respect to the periodic boundary condition. By exploiting Green functions, eigenvalues, rotation numbers, and their estimates, we will give several optimal conditions.