  A curious feature of complex scattering potentials v(x) is the appearance of
spectral singularities. We offer a quantitative description of spectral
singularities that identifies them with an obstruction to the existence of a
complete biorthonormal system consisting of the eigenfunctions of the
Hamiltonian operator, i.e., -\frac{d^2}{dx^2}+v(x), and its adjoint. We
establish the equivalence of this description with the mathematicians'
definition of spectral singularities for the potential
v(x)=z_-\delta(x+a)+z_+\delta(x-a), where z_\pm and a are respectively complex
and real parameters and \delta(x) is the Dirac delta-function. We offer a
through analysis of the spectral properties of this potential and determine the
regions in the space of the coupling constants z_\pm where it admits bound
states and spectral singularities. In particular, we find an explicit bound on
the size of certain regions in which the Hamiltonian is quasi-Hermitian and
examine the consequences of imposing PT-symmetry.
