Non-Hermitian quantum Fermi accelerator

We exactly solve a quantum Fermi accelerator model consisting of a time-independent non-Hermitian Hamiltonian with time-dependent Dirichlet boundary conditions. A Hilbert space for such systems can be defined in two equivalent ways, either by first constructing a time-independent Dyson map and subsequently unitarily mapping to fixed boundary conditions or by first unitarily mapping to fixed boundary conditions followed by the construction of a time-dependent Dyson map. In turn, this allows to construct time-dependent metric operators from a time-independent metric and two time-dependent unitary maps that freeze the moving boundaries. From the time-dependent energy spectrum, we find the known possibility of oscillatory behavior in the average energy in the PT regime, whereas in the spontaneously broken PT regime we observe the feature of a one-time depletion of the energy. We show that the PT broken regime is mended with a moving boundary, equivalently to mending it with a time-dependent Dyson map.