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On the absolutely continuous spectrum of one-dimensional quasi-periodic Schrödinger operators in the adiabatic limit

Authors: Alexander Fedotov and Frédéric Klopp
Journal: Trans. Amer. Math. Soc. 357 (2005), 4481-4516
MSC (2000): Primary 34L40, 34E20, 81Q05, 81Q20
Published electronically: June 21, 2005
MathSciNet review: 2156718
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Abstract: In this paper we study the spectral properties of families of quasi-periodic Schrödinger operators on the real line in the adiabatic limit in the case when the adiabatic iso-energetic curves are extended along the position direction. We prove that, in energy intervals where this is the case, most of the spectrum is purely absolutely continuous in the adiabatic limit, and that the associated generalized eigenfunctions are Bloch-Floquet solutions.

RÉSUMÉ. Cet article est consacré à l'étude du spectre de certaines familles d'équations de Schrödinger quasi-périodiques sur l'axe réel lorsque les variétés iso-énergetiques adiabatiques sont étendues dans la direction des positions. Nous démontrons que, dans un intervalle d'énergie où ceci est le cas, le spectre est dans sa majeure partie purement absolument continu et que les fonctions propres généralisées correspondantes sont des fonctions de Bloch-Floquet.

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Additional Information

Alexander Fedotov
Affiliation: Department of Mathematical Physics, St. Petersburg State University, 1, Ulianovskaja, 198904 St. Petersburg-Petrodvorets, Russia

Frédéric Klopp
Affiliation: Département de Mathématique, Institut Galilée, U.R.A 7539 C.N.R.S, Université de Paris-Nord, Avenue J.-B. Clément, F-93430 Villetaneuse, France

Keywords: Quasi-periodic Schr{\"o}dinger equation, absolutely continuous spectrum, Bloch-Floquet solutions, complex WKB method, monodromy matrix, adiabatic limit
Received by editor(s): November 14, 2003
Published electronically: June 21, 2005
Additional Notes: This work was done while the first author held a PAST professorship at Université Paris 13. The second author gratefully acknowledges support of the European TMR network ERBFMRXCT960001.
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.