Lattice Kronig-Penney Models

We discuss periodic Schrödinger operators for a particle on a rectangular lattice of sides ${\ell }_1,{\ell }_2$. In addition to the standard ( $\delta$-type) coupling with continuous wave functions at lattice nodes, we introduce two other boundary conditions which generalize naturally the one-dimensional ${\delta }^{\prime }$ 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 $\delta$ lattice has no gaps above the threshold if ${\ell }_2/{\ell }_1$ is badly approximable by rationals and the coupling constant is small enough.