Exact analytical solutions of two Hill’s equations that have continuously differentiable coefficients are obtained in terms of the Floquet–Bloch fundamental system. New features of the band structures of those equations are reported and investigated.
Caffrey, S., Morozov, G. V., MacBeath, D., & Sprung, D. W. L. (2016). Floquet-Bloch analysis of analytically solvable Hill equations with continuous potentials. Journal of the Optical Society of America B, 33(6), 1190-1196. https://doi.org/10.1364/JOSAB.33.001190