  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 new feature of a one-time depletion of the energy. We show that the
PT broken regime is mended with moving boundary, equivalently to mending it
with a time-dependent Dyson map.
