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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
Posted:
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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Reducibility or non-uniform hyperbolicity for quasiperiodic Schrödinger cocycles,
to appear in Annals of Mathematics.
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- J. Bellissard, R. Lima, and D. Testard.
Metal-insulator transition for the Almost Mathieu model.
Communications in Mathematical Physics, 88:207-234, 1983. MR 0696805 (85i:82055)
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- V. Buslaev.
On spectral properties of adiabatically perturbed Schrödinger operators with periodic potentials.
In Séminaires d'équations aux dérivées partielles, volume XVIII, Palaiseau, 1991.
Ecole Polytechnique. MR 1131596 (93g:35103)
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The complex WKB method for Harper's equation.
Preprint, Mittag-Leffler Institute, Stockholm, 1993. MR 1301830 (96f:34005)
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- V. Buslaev and A. Fedotov.
Monodromization and Harper equation.
In: Equations aux dérivées partielles, 1994,
Ecole Polytechnique, Paris, France. MR 1300917 (95i:81043)
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- V. Buslaev and A. Fedotov.
Bloch solutions of difference equations.
St. Petersburg Math. Journal, 7:561-594, 1996. MR 1356532 (96m:47060)
- 7.
- E. I. Dinaburg and Ja. G. Sina
 .
The one-dimensional Schrödinger equation with quasiperiodic potential.
Funkcional. Anal. i Prilozen., 9(4):8-21, 1975. MR 0470318 (57:10076)
- 8.
- M. Eastham.
The spectral theory of periodic differential operators.
Scottish Academic Press, Edinburgh, 1973.
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- L. H. Eliasson.
Floquet solutions for the 1-dimensional quasi-periodic Schrödinger equation.
Communications in Mathematical Physics, 146:447-482, 1992. MR 1167299 (93d:34141)
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- A. Fedotov and F. Klopp.
A complex WKB analysis for adiabatic problems.
Asymptotic Analysis, 27:219-264, 2001. MR 1858917 (2002h:81069)
- 11.
- A. Fedotov and F. Klopp.
Anderson transitions for a family of almost periodic Schrödinger equations in the adiabatic case.
Comm. Math. Phys., 227(1):1-92, 2002. MR 1903839 (2003f:82043)
- 12.
- V. Marchenko and I. Ostrovskii.
A characterization of the spectrum of Hill's equation.
Math. USSR Sbornik, 26:493-554, 1975.
- 13.
- H. McKean and P. van Moerbeke.
The spectrum of Hill's equation.
Inventiones Mathematicae, 30:217-274, 1975. MR 0397076 (53:936)
- 14.
- L. Pastur and A. Figotin.
Spectra of Random and Almost-Periodic Operators.
Springer-Verlag, Berlin, 1992. MR 1223779 (94h:47068)
- 15.
- E.C. Titchmarsh.
Eigenfunction expansions associated with second-order differential equations. Part II.
Clarendon Press, Oxford, 1958. MR 0094551 (20:1065)
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- M. Wilkinson.
Critical properties of electron eigenstates in incommensurate systems.
Proc. Roy. Soc. London Ser. A, 391(1801):305-350, 1984. MR 0739684 (86b:81136)
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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:
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
DOI:
http://dx.doi.org/10.1090/S0002-9947-05-03961-9
PII:
S 0002-9947(05)03961-9
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
Posted:
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.
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