A Schrödinger operator with a sparse potentials models a system where you have an impurity every now and then, but less and less frequently as you move out to infinity. Such operators are very well understood if the distance between consecutive impurities grows very fast. For moderate growth and (let's say) identical impuritites I don't know of any general result on the type of the spectrum. It is expected that a rich variety of phenomena can occur for such systems.