JournalsjfgVol. 4 , No. 4pp. 329–337

Discrete Schrödinger operators with potentials defined by measurable sampling functions over Liouville torus rotations

  • Alexander Y. Gordon

    University of North Carolina at Charlotte, USA
Discrete Schrödinger operators with potentials defined by measurable sampling functions over Liouville torus rotations cover
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Abstract

We consider a one-dimensional discrete Schrödinger operator Hω=Δ+vωH_\omega=\Delta+v_\omega whose potential vωv_\omega has the form vω(j)=V(ω+jαˉ),jZv_\omega(j)=V(\omega+j\bar{\alpha}), j\in\mathbb{Z}. Here ωTk=Rk/Zk, αRk,  αˉ\omega\in\mathbb{T}^k=\mathbb{R}^k/\mathbb{Z}^k,\ \alpha\in\mathbb{R}^k,\ \ \bar{\alpha} is the projection of α\alpha on Tk\mathbb{T}^k, and the function V ⁣:TkCV\colon \mathbb{T}^k\to\mathbb{C} is Borel measurable. We show that if the frequency vector α\alpha is Liouville (the sequence {ναˉ}νN\{\nu \bar{\alpha}\}_{\nu\in\mathbb N} has a subsequence converging to 00 fast enough), then for Lebesgue almost every ωTk\omega\in\mathbb T^k the point spectrum of the operator HωH_\omega is empty.

Cite this article

Alexander Y. Gordon, Discrete Schrödinger operators with potentials defined by measurable sampling functions over Liouville torus rotations. J. Fractal Geom. 4 (2017), no. 4 pp. 329–337

DOI 10.4171/JFG/53