Advances in Mathematical Physics
Volume 2010 (2010), Article ID 145436, 30 pages
doi:10.1155/2010/145436
Research Article

On the Singular Spectrum for Adiabatic Quasiperiodic Schrödinger Operators

Magali Marx1 and Hatem Najar2

1LAGA, U.M.R. 7539 C.N.R.S, Institut Galilée, Université de Paris-Nord, 99 Avenue J.-B. Clément, 93430 Villetaneuse, France
2Département de Mathématiques, I.S.M.A.I. Kairouan, Abd Assed Ibn Elfourat, 3100 Kairouan, Tunisia

Received 25 June 2009; Revised 28 November 2009; Accepted 28 February 2010

Academic Editor: Pavel Kurasov

Copyright © 2010 Magali Marx and Hatem Najar. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We study spectral properties of a family of quasiperiodic Schrödinger operators on the real line in the adiabatic limit. We assume that the adiabatic iso-energetic curve has a real branch that is extended along the momentum direction. In the energy intervals where this happens, we obtain an asymptotic formula for the Lyapunov exponent and show that the spectrum is purely singular. This result was conjectured and proved in a particular case by Fedotov and Klopp (2005).