1D periodic potentials with gaps vanishing at k=0

TL;DR

This paper characterizes one-dimensional periodic potentials where all energy gaps vanish at the Brillouin zone center, revealing their mathematical structure and relevance across various physics fields.

Contribution

It provides a necessary and sufficient condition for such potentials, connecting quantum mechanics, differential equations, and physical applications.

Findings

All gaps vanish at k=0 for these potentials.

Characterization through a specific mathematical condition.

Relevance to supersymmetric QM, Korteweg-de Vries, and diffusion.

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 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.