Delone measures of finite local complexity and applications to spectral theory of one-dimensional continuum models of quasicrystals

Steffen Klassert; Daniel Lenz; Peter Stollmann

doi:10.3934/dcds.2011.29.1553

Article Contents

2011, Volume 29, Issue 4: 1553-1571. Doi: 10.3934/dcds.2011.29.1553

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Delone measures of finite local complexity and applications to spectral theory of one-dimensional continuum models of quasicrystals

1.
Fakultät für Mathematik, TU Chemnitz, D - 09107 Chemnitz
2.
Mathematisches Institut, Friedrich-Schiller Universität, Ernst-Abbe-Platz 2, D - 07743 Jena

Received: December 31, 2009

Revised: August 31, 2010

Published: September 2011

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Abstract

We study measures on the real line and present various versions of what it means for such a measure to take only finitely many values. We then study perturbations of the Laplacian by such measures. Using Kotani-Remling theory, we show that the resulting operators have empty absolutely continuous spectrum if the measures are not periodic. When combined with Gordon type arguments this allows us to prove purely singular continuous spectrum for some continuum models of quasicrystals.

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Mathematics Subject Classification: Primary:81Q10, 35J10; Secondary: 52C32, 82D30.

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