Non-Hermitian quantum mechanics is a theory that deals with two types of physical phenomena. One type of phenomena cannot be described by the standard (Hermitian) quantum mechanics since the local potentials in the hamiltonians are complex. The second type of phenomena are associated with local real potentials that support continuous spectra.
The second type of phenonena can be described only by the time dependent Schrödinger equation. The potentials can be complex because of different reasons, such as when complex absorbing potentials (CAPs) are introduced into the physical Hamiltonian in order to enable one to carry out long time wavepacket propagations by using finite grids or finte number of basis functions that impose box quantization boundary conditions on the solutions on the time-dependent and the time independent Schroedinger equations. The reflection free CAPs RFCAP suppress the artificial reflections of the tail of the wavepackets from the edge of the grid or from the use of box-quantization boundary conditions.