LATTICE KRONIG-PENNEY MODELS

We discuss periodic Schrödinger operators for a particle on a rectangular lattice of sides 1, 2. In addition to the standard (-type) coupling with continuous wave functions at lattice nodes, we introduce two other boundary conditions which generalize naturally the one-dimensional interaction and its symmetrized version; both of them can be used as models for geometric scatterers. We show that the band spectrum of these models depends on number-theoretic properties of the parameters. In particular, the lattice has no gaps above the threshold if 2/1 is badly approximable by rationals and the coupling constant is small enough. © 1995 The American Physical Society.