Abstract
We studyH=−d 2/dx 2+V(x) withV(x) limit periodic, e.g.V(x)=Σa n cos(x/2n) with Σ∣a n ∣<∞. We prove that for a genericV (and for generica n in the explicit example), σ(H) is a Cantor (≡ nowhere dense, perfect) set. For a dense set, the spectrum is both Cantor and purely absolutely continuous and therefore purely recurrent absolutely continuous.
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Communicated by J. Ginibre
Research partially supported by NSF Grant MCS78-01885
On leave from Department of Physics, Princeton University
On leave from Departments of Mathematics and Physics, Princeton University; during 1980–81 Sherman Fairchild Visiting Scholar at Caltech
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Avron, J., Simon, B. Almost periodic Schrödinger operators. Commun.Math. Phys. 82, 101–120 (1981). https://doi.org/10.1007/BF01206947
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DOI: https://doi.org/10.1007/BF01206947