O. Zagordi and A. Michelangeli

1D Periodic Potentials with Gaps Vanishing at $k=0$

abstract:
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 characterise 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 occurs of linearly independent solutions of the corresponding Schrödinger equation (Hill's equation). This result is placed in the perspective of the previous related results available in the literature.

Mathematics Subject Classification: 30RD10, 30RD15, 30RD20, 34B24, 34B30, 34E05, 34L05, 34L99, 46N20, 46N50, 47N20, 47N50, 81Q10, 81V45

Key words and phrases: Schrödinger equation with periodic potential, dispersion relations, energy bands and gaps, vanishing gaps, Hill's equation, intervals of stability and instability, discriminant and characteristic values of an O.D.E., coexistence.