  Appearance of energy bands and gaps in the dispersion relations of a periodic
potential is a standard feature of Quantum Mechanics. We investigate the class
of one-dimensional periodic potentials for which all gaps vanish at the center
of the Brillouin zone. We characterize them through a necessary and sufficient
condition. Potentials of the form we focus on arise in different fields of
Physics, from supersymmetric Quantum Mechanics, to Korteweg-de Vries equation
theory and classical diffusion problems. The O.D.E. counterpart to this problem
is the characterisation of periodic potentials for which coexistence occur of
linearly independent solutions of the corresponding Schroedinger equation
(Hill's equation). This result is placed in perspective of the previous related
results available in the literature.
