$PT$-symmetric dynamical confinement: Fermi acceleration, quantum force, and Berry phase

We consider a quantum particle under the dynamical confinement caused by $PT$-symmetric box with a moving wall. The latter is described in terms of the time-dependent Schrödinger equation obeying the time-dependent $PT$-symmetric boundary conditions. The class of the functions, describing time dependence of the wall's position and keeping the system as $PT$-symmetric is found. Physically observable characteristics, such as average kinetic energy and the average quantum force are calculated as a function of time. It is found that the behavior of the average kinetic energy as a function of time is completely different than that for of Hermitian counterpart of the model, while the average quantum force behaves similarly to that for Hermitian system. Also, geometric phase is calculated for the harmonically oscillating wall regime. The experimental realization of the proposed model is discussed.