General solution theory for Schrödinger's equation in arbitrary 2D-periodic spatial structures I. The monolayer problem

Abstract

The electronic states in arbitrary spatial structures exhibiting 2-dimensional translational symmetry are most efficiently determined within a 3-step concept: First, for any single monolayer the structure is composed of, the (physically relevant) solutions to Schrödinger's equation are characterized and parametrized by means of appropriately chosen boundary data. Then, the original structure is synthesized by applying the “layer composition process,” and finally the electronic wave functions describing the underlying physical situation in the whole structure are obtained by adjusting them to the respective asymptotics (assembly of boundary controlled monolayers method). Envisaging the monolayer problem in this paper, the existence and uniqueness of both weak and strong solutions is shown. While weak solutions lead to a variational characterization, which is useful for estimating the quality of numerical approximations, the conception of strong solutions yields a realization of the monolayer Schrödinger operator in the Hilbert space L 2, the spectral properties of which are not only of practical interest (calculation of band structures) but also prove to be crucial for the convergence of the layer composition process (as will be shown in a subsequent article).