International Journal of Difference Equations

We first give a short review on supermodels in quantum mechanics where the main focus will be on selfsimilar supermodels, showing a point spectrum which consists of basic q-versions of the natural numbers. We want to relate these supermodels to discrete versions of Schrödinger operators which exhibit — at least partially — the same type of discrete point spectrum. To do so, the concept of strip discretizations is reviewed on basic linear grids. This type of discretization shows the typical point spectrum, consisting of basic q-versions of the natural numbers. Precisely the same type of spectrum is finally also presented in case of basic multigrid discretizations. We therefore obtain a unified discrete model of some Schrödinger equations which allow both, piecewise continuous solutions (Section 4) and sophisticated multigrid solutions (Section 5) — a scenario which plays already a great role in approaches established by A. Lorek and B. Heim.