  We consider the eigenvalue problem for one-dimensional linear Schr\"odinger
lattices (tight-binding) with an embedded few-sites linear or nonlinear,
Hamiltonian or non-conservative defect (an oligomer). Such a problem arises
when considering scattering states in the presence of (generally complex)
impurities as well as in the stability analysis of nonlinear waves. We describe
a general approach based on a matching of solutions of the linear portions of
the lattice at the location of the oligomer defect. As specific examples we
discuss both linear and nonlinear, Hamiltonian and $\cP \cT$-symmetric dimers
and trimers. In the linear case, this approach provides us a handle for
semi-analytically computing the spectrum [this amounts to the solution of a
polynomial equation]. In the nonlinear case, it enables the computation of the
linearization spectrum around the stationary solutions. The calculations
showcase the oscillatory instabilities that strongly nonlinear states typically
manifest.
