Periodic Schrödinger operators with large gaps and Wannier-Stark ladders

Abstract

We describe periodic, one dimensional Schr\"odinger operators, with the property that the widths of the forbidden gaps increase at large energies and the gap to band ratio is not small. Such systems can be realized by periodic arrays of geometric scatterers, e.g., a necklace of rings. Small, multichannel scatterers lead (for low energies) to the same band spectrum as that of a periodic array of (singular) point interactions known as \ensuremath{\delta}'. We consider the Wannier-Stark ladder of \ensuremath{\delta}' and show that the corresponding Schr\"odinger operator has no absolutely continuous spectrum.