  We address static and dynamical properties of one-dimensional (1D) quantum
droplets (QDs) under the action of local potentials in the form of narrow wells
and barriers. The QDs are governed by the 1D Gross-Pitaevskii equation
including the mean-field cubic repulsive term and the beyond-mean-field
attractive quadratic one. In the case of the well represented by the
delta-functional potential, three exact stable solutions are found for
localized states pinned to the well. The Thomas-Fermi approximation for the
well and the adiabatic approximation for the collision of the QD with the
barrier are developed too. Collisions of incident QDs with the wells and
barriers are analyzed in detail by means of systematic simulations. Outcomes,
such as fission of the moving QD into transmitted, reflected, and trapped
fragments, are identified in relevant parameter planes. In particular, a
counter-intuitive effect of partial or full rebound of the incident QD from the
potential well is studied in detail and qualitatively explained.
