- 00-217 Dirk Hundertmark, Werner Kirsch
- Spectral theory of sparse potentials
May 9, 00
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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
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".