Our goal is to show that large classes of Schro dinger operators H=&2+V in L 2 (R d ) exhibit intervals of dense pure point spectrum, in any dimension d. We approach this by assuming that the potential V(x) coincides with a potential V 0 (x) of a comparison operator'' H 0 =&2+V 0 on a sequence of ring shaped (but nog necessarily spherical) regions U n , n=1, 2, ... . For energies in the resolvent set \(H 0 ) of H 0 the regions U n act as effective barriers'' in the sense of quantum mechanical scattering under the potential V. Under certain assumptions on the geometry of the U n and their complements we show that (i)
Here _ ac and _ c denote the absolutely continuous and continuous spectrum, respectively, and H(*) is a ``local randomization'' of H, i.e., H(*)=H+*W, where W is any continuous and compactly supported perturbation of fixed sign. Our assumptions leave plenty of room for examples where the spectrum of H fills entire spectral gaps of H 0 . This leads to intervals of dense pure point spectrum for H(*). We also give an explicit decay estimate for eigenfunctions, thus establishing localization for H(*) in arbitrary spectral gaps of H 0 .
1997 Academic Press
1. THE RESULTS
The entry of questions from the physics of disordered media has considerably changed and enriched the spectral theory of Schro dinger operators in the last two decades. It has been found that potentials with random, quasiperiodic, or other irregular types of potential asymptotics lead to ``non-classical'' spectral phenomena like localization (a notion coming from physics and now generally accepted to mean dense pure point spectrum with exponentially decaying eigenfunctions) or singular continuous spectrum. This shift of attention is stressed, for example, in the recent review [15]. For general information on random and quasiperiodic potentials see the books [13] and [1].
One-dimensional disordered media are now well understood from the viewpoint of spectral theory, but fundamental questions remain open in article no.