General solution theory for Schrödinger's equation in arbitrary 2D-periodic spatial structures II. The synthesis of global solutions by layer composition

After the first step in the assembly of boundary controlled monolayers method, namely the monolayer boundary value problem, was dealt with in a preceding article, here the synthesis of the original 2D-periodic spatial structure by means of the "layer composition process" is investigated. Since the convergence properties of this process are most appropriately discussed in terms of generalized Bloch waves, these are first analyzed in full detail; special interest is here paid to a "current sum rule," which proves to be fundamental for characterizing the asymptotic behavior of the reflection and transmission operators of the synthesized structure. As an illustrating example, it is finally shown how to calculate global solutions to Schrödinger's equation for a semi-infinite crystal structure (with real physical surface) which match the correct asymptotic boundary data in both the vacuum region and the interior of the crystal deeply in the bulk.