00-217 Dirk Hundertmark, Werner Kirsch

Spectral theory of sparse potentials (94K, LaTeX2e) 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 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".

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