Sub-Planck scale structures in the P{\"o}schl-Teller potential and their sensitivity to perturbations

We find the existence of sub-Planck scale structures in the P{\"o}schl-Teller potential, which is an exactly solvable potential with both symmetric and asymmetric features. We analyze these structures in both cases by looking at the Wigner distribution of the state evolved from an initial coherent state up to various fractional revival times. We also investigate the sensitivity to perturbations of the P{\"o}schl-Teller potential and we verify that, similar to the harmonic oscillator, the presence of sub-Planck structure in phase space is responsible for a high sensitivity to phase-space displacements.