  We consider a quantum particle, moving on a lattice with a tight-binding
Hamiltonian, which is subjected to measurements to detect it's arrival at a
particular chosen set of sites. The projective measurements are made at regular
time intervals $\tau$, and we consider the evolution of the wave function till
the time a detection occurs. We study the probabilities of its first detection
at some time and conversely the probability of it not being detected (i.e.,
surviving) up to that time. We propose a general perturbative approach for
understanding the dynamics which maps the evolution operator, consisting of
unitary transformations followed by projections, to one described by a
non-Hermitian Hamiltonian. For some examples, of a particle moving on one and
two-dimensional lattices with one or more detection sites, we use this approach
to find exact expressions for the survival probability and find excellent
agreement with direct numerical results. A mean field model with hopping
between all pairs of sites and detection at one site is solved exactly. For the
one- and two-dimensional systems, the survival probability is shown to have a
power-law decay with time, where the power depends on the initial position of
the particle. Finally, we show an interesting and non-trivial connection
between the dynamics of the particle in our model and the evolution of a
particle under a non-Hermitian Hamiltonian with a large absorbing potential at
some sites.
