  Spectral singularities are predicted to occur in a non-Hermitian extension of
the Friedrichs-Fano-Anderson model describing the decay of a discrete state $|a
>$ coupled to a continuum of modes. A physical realization of the model, based
on electronic or photonic transport in a semi-infinite tight-binding lattice
with an imaginary impurity site at the lattice boundary, is proposed. The
occurrence of the spectral singularities is shown to correspond either to a
diverging reflection probability (for an amplifying impurity) or to a vanishing
reflection probability (for an absorbing impurity) from the lattice boundary.
In the former case, the spectral singularity of the resolvent is also
responsible for the non-decay of state $|a>$ into the continuum, in spite of
the absence of bound states.
