Skip to Main Content

Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality 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 1.43.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

On the absolutely continuous spectrum of one-dimensional quasi-periodic Schrödinger operators in the adiabatic limit
HTML articles powered by AMS MathViewer

by Alexander Fedotov and Frédéric Klopp PDF
Trans. Amer. Math. Soc. 357 (2005), 4481-4516 Request permission

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.

References
  • A. Avila, and R. Krikorian. Reducibility or non-uniform hyperbolicity for quasiperiodic Schrödinger cocycles, to appear in Annals of Mathematics.
  • J. Bellissard, R. Lima, and D. Testard, A metal-insulator transition for the almost Mathieu model, Comm. Math. Phys. 88 (1983), no. 2, 207–234. MR 696805, DOI 10.1007/BF01209477
  • V. Buslaev, On spectral properties of adiabatically perturbed Schroedinger operators with periodic potentials, Séminaire sur les Équations aux Dérivées Partielles, 1990–1991, École Polytech., Palaiseau, 1991, pp. Exp. No. XXIII, 15. MR 1131596
  • V. S. Buslaev and A. A. Fedotov, The complex WKB method for the Harper equation, Algebra i Analiz 6 (1994), no. 3, 59–83 (Russian); English transl., St. Petersburg Math. J. 6 (1995), no. 3, 495–517. MR 1301830
  • V. Buslaev and A. Fedotov, The monodromization and Harper equation, Séminaire sur les Équations aux Dérivées Partielles, 1993–1994, École Polytech., Palaiseau, 1994, pp. Exp. No. XXI, 23. MR 1300917
  • V. S. Buslaev and A. A. Fedotov, Bloch solutions for difference equations, Algebra i Analiz 7 (1995), no. 4, 74–122 (Russian); English transl., St. Petersburg Math. J. 7 (1996), no. 4, 561–594. MR 1356532
  • E. I. Dinaburg and Ja. G. Sinaĭ, The one-dimensional Schrödinger equation with quasiperiodic potential, Funkcional. Anal. i Priložen. 9 (1975), no. 4, 8–21 (Russian). MR 0470318
  • M. Eastham. The spectral theory of periodic differential operators. Scottish Academic Press, Edinburgh, 1973.
  • L. H. Eliasson, Floquet solutions for the $1$-dimensional quasi-periodic Schrödinger equation, Comm. Math. Phys. 146 (1992), no. 3, 447–482. MR 1167299, DOI 10.1007/BF02097013
  • Alexander Fedotov and Frédéric Klopp, A complex WKB method for adiabatic problems, Asymptot. Anal. 27 (2001), no. 3-4, 219–264 (English, with English and French summaries). MR 1858917
  • Alexander Fedotov and Frédéric Klopp, Anderson transitions for a family of almost periodic Schrödinger equations in the adiabatic case, Comm. Math. Phys. 227 (2002), no. 1, 1–92 (English, with English and French summaries). MR 1903839, DOI 10.1007/s002200200612
  • V. Marchenko and I. Ostrovskii. A characterization of the spectrum of Hill’s equation. Math. USSR Sbornik, 26:493–554, 1975.
  • H. P. McKean and P. van Moerbeke, The spectrum of Hill’s equation, Invent. Math. 30 (1975), no. 3, 217–274. MR 397076, DOI 10.1007/BF01425567
  • Leonid Pastur and Alexander Figotin, Spectra of random and almost-periodic operators, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 297, Springer-Verlag, Berlin, 1992. MR 1223779, DOI 10.1007/978-3-642-74346-7
  • E. C. Titchmarsh, Eigenfunction expansions associated with second-order differential equations. Vol. 2, Oxford, at the Clarendon Press, 1958. MR 0094551, DOI 10.1063/1.3062231
  • M. Wilkinson, Critical properties of electron eigenstates in incommensurate systems, Proc. Roy. Soc. London Ser. A 391 (1984), no. 1801, 305–350. MR 739684
Similar Articles
Additional Information
  • Alexander Fedotov
  • Affiliation: Department of Mathematical Physics, St. Petersburg State University, 1, Ulianovskaja, 198904 St. Petersburg-Petrodvorets, Russia
  • Email: fedotov@mph.phys.spbu.ru
  • 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: klopp@math.univ-paris13.fr
  • 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: https://doi.org/10.1090/S0002-9947-05-03961-9
  • MathSciNet review: 2156718