Analysis and Discussion of Solutions to the Schrödinger Equation Under a Sawtooth Potential

Sawtooth potentials-piecewise-linear potentials defined by alternating rising and falling slopes-have garnered interest for their roles in classical Brownian ratchets, quantum transport, and flat bands in ultracold-atom lattices.However, a general solution of the time-independent Schrödinger equation for an arbitrary asymmetric sawtooth potential and a study of its band structure remain absent from the literature.We derive the exact analytical eigenstates in terms of Airy functions, but due to numerical difficulty, we solved for the eigenstates using a piecewise constant "staircase approximation."We find localized low-energy states confined within individual wells and plane waves for high energies.We apply the obtained solutions to investigate the relationship between bandwidth and parameters of the sawtooth potentials, which could offer design principles for realizing flatbands in condensed-matter and ultracold-atom research.We find that the bandwidth of the lowest energy band is entirely independent of the sawtooth's asymmetry, while it remains inversely correlated with potential height and cell size.The methodologies presented here provide a toolkit for further exploration of sawtooth potentials.