  We introduce the projected Green's function technique to study quasi-periodic
systems such as the Andre-Aubry-Harper (AAH) model and beyond. In particular,
we use projected Green's functions to construct a "rational approximate"
sequence of transfer matrix equations consistent with quasi-periodic topology,
where convergence of these sequences corresponds to the existence of extended
eigenfunctions. We motivate this framework by applying it to a few well studied
cases such as the almost-Mathieu operator (AAH model), as well as more generic
non-dual models that challenge standard routines. The technique is flexible and
can be used to extract both analytic and numerical results, e.g. we
analytically extract a modified phase diagram for Liouville irrationals. As a
numerical tool, it does not require the fixing of boundary conditions and
circumvents a primary failing of numerical techniques in quasi-periodic
systems, extrapolation from finite size. Instead, it uses finite size scaling
to define convergence bounds on the full irrational limit.
