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Dynamics of the Quasi-Periodic Schrödinger Cocycle at the Lowest Energy in the Spectrum

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Abstract

In this paper we consider the quasi-periodic Schrödinger cocycle over \(\mathbb{T}^d\) (d ≥ 1) and, in particular, its projectivization. In the regime of large coupling constants and Diophantine frequencies, we give an affirmative answer to a question posed by M. Herman [21, p.482] concerning the geometric structure of certain Strange Non-chaotic Attractors which appear in the projective dynamical system. We also show that for some phase, the lowest energy in the spectrum of the associated Schrödinger operator is an eigenvalue with an exponentially decaying eigenfunction. This generalizes [39] to the multi-frequency case (d > 1).

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Authors and Affiliations

  1. Department of Mathematics, University of Toronto, Toronto, Ontario, Canada, M5S 2E4

    Kristian Bjerklöv

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  1. Kristian Bjerklöv
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Correspondence to Kristian Bjerklöv.

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Communicated by G. Gallavotti

Research partially supported by STINT (Institutional Grant 2002-2052), The Royal Swedish Academy of Sciences and SVeFUM

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Bjerklöv, K. Dynamics of the Quasi-Periodic Schrödinger Cocycle at the Lowest Energy in the Spectrum. Commun. Math. Phys. 272, 397–442 (2007). https://doi.org/10.1007/s00220-007-0238-y

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