Abstract
The criterion for vanishing or zero energy gaps in the spectrum of 1D electronic bandstructures is obtained from perturbation theory. It is in agreement with examples of periodic potentials which never exhibit zero-gaps as well as with those which may, including certain smooth periodic potentials. The results have relevance to recent work in the field of semiconductor layered heterostructures or superlattices. The analysis should also have implications for other problems that involve Hill's determinant.