  In recent work, the so-called quasi-Zeno dynamics of a system has been
investigated in the context of the quantum first passage problem. This dynamics
considers the time evolution of a system subjected to a sequence of selective
projective measurements made at small but finite intervals of time. This means
that one has a sequence of steps, with each step consisting of a unitary
transformation followed by a projection. The dynamics is non-unitary and, in
recent work, it has been shown that it can be effectively described by two
different non-Hermitian Hamiltonians. Here we explore this connection by
considering the problem of detecting a free quantum particle moving on a
one-dimensional lattice, where the detector is placed at the origin and the
particle is initially located at some specified lattice point. We find that
results for distribution times for the first detection probability, obtained
from the non-Hermitian Hamiltonians, are in excellent agreement with known
exact results as well as exact numerics. Interesting finite-size effects are
discussed. We also study the first detection problem for the example of a
particle moving in a quasi-periodic potential, an example where the unperturbed
particle's motion can be ballistic, localized or diffusive.
