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On the structure of spectra of periodic elliptic operators


Authors: Peter Kuchment and Sergei Levendorskiî
Journal: Trans. Amer. Math. Soc. 354 (2002), 537-569
MSC (2000): Primary 35P99; Secondary 35J10
DOI: https://doi.org/10.1090/S0002-9947-01-02878-1
Published electronically: September 21, 2001
MathSciNet review: 1862558
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Abstract: The paper discusses the problem of absolute continuity of spectra of periodic elliptic operators. A new result on absolute continuity for a matrix operator of Schrödinger type is obtained. It is shown that all types of operators for which the absolute continuity has previously been established can be reduced to this one. It is also discovered that some natural generalizations stumble upon an obstacle in the form of non-triviality of a certain analytic bundle on the two-dimensional torus.


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Additional Information

Peter Kuchment
Affiliation: Department of Mathematics and Statistics, Wichita State University, Wichita, Kansas 67260-0033
Address at time of publication: Department of Mathematics, Texas A&M University, College Station, Texas 77843-3368
Email: kuchment@math.tamu.edu

Sergei Levendorskiî
Affiliation: Department of Mathematics and Statistics, Wichita State University, Wichita, Kansas 67260-0033
Address at time of publication: Department of Mathematics, Rostov State Academy of Economics, Rostov-on-Don, Russia
Email: leven@ns.rnd.runnet.ru

DOI: https://doi.org/10.1090/S0002-9947-01-02878-1
Keywords: Periodic operator, elliptic operator, absolutely continuous spectrum, Schr\"{o}dinger operator, magnetic and electric potentials
Received by editor(s): October 3, 2000
Published electronically: September 21, 2001
Additional Notes: The first author was supported in part by an NRC COBASE Grant, NSF Grants DMS 9610444 and DMS 0072248, and by a DEPSCoR Grant
The second author was supported in part by an NRC COBASE Grant.
Article copyright: © Copyright 2001 American Mathematical Society

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