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

We present general solutions to the Schrodinger equation for a di-periodic potential composed of two frequencies, thus generalizing the standard sinusoidal potential. The Schrodinger 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. (C) 2021 Optical Society of America

Automated summary

The paper presents general solutions to the Schrodinger equation for a di-periodic potential, extending it from the standard sinusoidal potential. Two different approaches were used to solve the three-term Whittaker-Hill equation with the di-periodic potential, resulting in the determination of Floquet-Bloch eigenwaves and band structure. Additionally, it was found that the Ince functions can describe the Mathieu functions obtained with purely sinusoidal potentials.