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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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On the absolutely continuous spectrum of one-dimensional quasi-periodic Schrödinger operators in the adiabatic limit
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by Alexander Fedotov and Frédéric Klopp PDF
Trans. Amer. Math. Soc. 357 (2005), 4481-4516 Request permission


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
  • Email:
  • 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
  • Email:
  • 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.
  • © Copyright 2005 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Trans. Amer. Math. Soc. 357 (2005), 4481-4516
  • MSC (2000): Primary 34L40, 34E20, 81Q05, 81Q20
  • DOI:
  • MathSciNet review: 2156718