Dirk Hundertmark, Werner Kirsch
Spectral theory of sparse potentials
(94K, LaTeX2e)

ABSTRACT.  We give a number of results concerning different possible spectral 
types for Schr\"odinger operators with sparse potentials. 
These potentials are in between stationary (e.g., random) 
potentials and the short range potentials familiar from 
scattering theory. They decay at infinity in some averaged sense, 
however in such a way that there is enough ``space" for surprising 
spectral properties. 
For a broad class of sparse potentials we establish existence of 
absolutely continuous spectrum above zero with scattering theory 
ideas. At the same time these potentials generically also possess 
negative essential spectrum. We classify this negative spectrum to 
some extend. It turns out to be pure point in many cases. In some 
cases the negative essential spectrum is countable, in fact it may 
be finite, but still does not belong to the discrete part of the 
spectrum. In other cases we find dense point spectrum. 
Finally, we treat "surface potentials" and prove analogous results 
for these "generalized sparse potentials".