Time-dependent optical potential for the Schrödinger solution in a truncated subspace

A time-dependent imaginary optical potential is used to approximate the Schr\"odinger time evolution in a necessarily truncated subspace. The flux of probability into the neglected space, for which the ordinary unitary subspace Schr\"odinger solution does not account, can generally be well reproduced by an optical potential that is diagonal in energy and linear in time. This is demonstrated in a number of examples including the Lipkin model, Gaussian Hamiltonians, and a schematic heavy-ion collision model.