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Journal of Spectral Theory


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Volume 6, Issue 3, 2016, pp. 557–600
DOI: 10.4171/JST/132

Published online: 2016-09-15

Localization for transversally periodic random potentials on binary trees

Richard Froese[1], Darrick Lee[2], Christian Sadel[3], Wolfgang Spitzer[4] and Günter Stolz[5]

(1) The University of British Columbia, Vancouver, Canada
(2) The University of British Columbia, Vancouver, Canada
(3) Pontificia Universidad Católica de Chile, Santiago de Chile, Chile
(4) FernUniversität Hagen, Germany
(5) University of Alabama at Birmingham, USA

We consider a random Schrödinger operator on the binary tree with a random potential which is the sum of a random radially symmetric potential, $Q_r$, and a random transversally periodic potential, $\kappa Q_t$, with coupling constant $\kappa$. Using a new one-dimensional dynamical systems approach combined with Jensen's inequality in hyperbolic space (our key estimate) we obtain a fractional moment estimate proving localization for small and large $\kappa$. Together with a previous result we therefore obtain a model with two Anderson transitions, from localization to delocalization and back to localization, when increasing $\kappa$. As a by-product we also have a partially new proof of one-dimensional Anderson localization at any disorder.

Keywords: Random Schrödinger operator, localization, Bethe lattice

Froese Richard, Lee Darrick, Sadel Christian, Spitzer Wolfgang, Stolz Günter: Localization for transversally periodic random potentials on binary trees. J. Spectr. Theory 6 (2016), 557-600. doi: 10.4171/JST/132