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Almost periodic Schrödinger operators: A Review

Simon, Barry (1982) Almost periodic Schrödinger operators: A Review. Advances in Applied Mathematics, 3 (4). pp. 463-490. ISSN 0196-8858. doi:10.1016/s0196-8858(82)80018-3.

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We review the recent rigorous literature on the one-dimensional Schrödinger equation, H = −d²/dx² + V(x) with V(x) almost periodic and the discrete (= tight binding) analog, i.e., the doubly infinite Jacobi matrix, h_(ij) = δ_(i,j+1) + δ_(i,j−1) + V_iδ_(i,j) with V_n almost periodic on the integers. Two themes dominate. The first is that the gaps in the spectrum tend to be dense, so that the spectrum is a Cantor set. We describe intuitions for this from the point of view of where gaps open, and from the point of view of anomalous long time behavior. We give a theorem of Avron and Simon, Chulaevsky, and Moser that for a generic sequence with Σ|a_n| < ∞, the continuum operator with V(x) = Σa_ncos(x/2^n) has a Cantor spectrum. The second theme involves unusual spectral types that tend to occur. We describe recurrent absolutely continuous spectrum, and show it occurs in some examples of the type just discussed. We give an intuition for dense point spectrum to occur, and some theorems on the occurrence of point spectrum. We sketch the proof of Avron and Simon, that for the discrete case with V_n = λ cos(2παn + ϑ), if λ > 2 and α is a Liouville number, then for a.e. ϑ, h has purely singular continuous spectrum.

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Simon, Barry0000-0003-2561-8539
Additional Information:© 1982 Published by Elsevier Under an Elsevier user license. Based, in part, on a talk at the VIth International Conference on Mathematical Physics, Berlin, August 11–20, 1981. Research partially supported by USNSF Grant MCS-78-01885. It is a pleasure to thank J. Bellisard, R. Johnson, J. Moser, and P. Sarnak for information on their work before publication, and to thank Gordon and Molchanov for discussions of Gordon's work. But most of all, I must express my gratitude to Yossi Avron; any insights in this paper are the result of a fruitful year of discussion, collaboration, and occasional argument.
Funding AgencyGrant Number
NSFMCS 78-01885
Issue or Number:4
Record Number:CaltechAUTHORS:20201007-081705850
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Official Citation:Barry Simon, Almost periodic Schrödinger operators: A Review, Advances in Applied Mathematics, Volume 3, Issue 4, 1982, Pages 463-490, ISSN 0196-8858,
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:105862
Deposited By: George Porter
Deposited On:07 Oct 2020 16:32
Last Modified:16 Nov 2021 18:47

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