  It is of interest in a variety of contexts, and in particular in the arrival
time problem, to consider the quantum state obtained through unitary evolution
of an initial state regularly interspersed with periodic projections onto the
positive $x$-axis (pulsed measurements). Echanobe, del Campo and Muga have
given a compelling but heuristic argument that the state thus obtained is
approximately equivalent to the state obtained by evolving in the presence of a
certain complex potential of step-function form. In this paper, with the help
of the path decomposition expansion of the associated propagators, we give a
detailed derivation of this approximate equivalence. The propagator for the
complex potential is known so the bulk of the derivation consists of an
approximate evaluation of the propagator for the free particle interspersed
with periodic position projections. This approximate equivalence may be used to
show that to produce significant reflection, the projections must act at time
spacing less than 1/E, where E is the energy scale of the initial state.
