The aim of this paper is to give several optimal criteria for the existence of maximum or antimaximum principles for the second order operator
, with periodic conditions, where
is a given periodic, integrable potential. This problem can be fully characterized in terms of periodic and antiperiodic eigenvalues. Alternatively, the maximum or antimaximum principle is closely related with the associated Green’s function and its sign. The author presents an illuminating and unifying review of this topic.