Floquet–Bloch eigenwaves and bandgaps in a di-periodic potential

Abstract

We present general solutions to the Schrödinger equation for a di-periodic potential composed of two frequencies, thus generalizing the standard sinusoidal potential. The Schrödinger equation with a di-periodic potential becomes a three-term Whittaker–Hill equation that is solved by two different approaches. The first is applying the central equation formalism, which allows determining the Floquet–Bloch eigenwaves and band structure as a function of the potential parameters. In the second approach, we transform the Whittaker–Hill equation into an Ince equation with a suitable change of variable. In this case, we get a complete set of orthogonal solutions described by Ince functions. The well-known Mathieu functions obtained with purely sinusoidal potentials are a special case of Ince functions.