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An overview of periodic elliptic operators


Author: Peter Kuchment
Journal: Bull. Amer. Math. Soc. 53 (2016), 343-414
MSC (2010): Primary 35B27, 35J10, 35J15, 35Q40, 35Q60, 47F05, 58J05, 81Q10; Secondary 32C99, 35C15, 47A53, 58J50, 78M40
DOI: https://doi.org/10.1090/bull/1528
Published electronically: April 6, 2016
MathSciNet review: 3501794
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Abstract: The article surveys the main topics, techniques, and results of the theory of periodic operators arising in mathematical physics and other areas. Close attention is paid to studying analytic properties of Bloch and Fermi varieties, which significantly influence most spectral features of such operators.

The approaches described are applicable not only to the standard model example of Schrödinger operator with periodic electric potential $ -\Delta +V(x)$, but to a wide variety of elliptic periodic equations and systems, equations on graphs, $ \overline {\partial }$-operator, and other operators on abelian coverings of compact bases.

Important applications are mentioned. However, due to the size restrictions, they are not dealt with in detail.


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Additional Information

Peter Kuchment
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843-3368
Email: kuchment@math.tamu.edu

DOI: https://doi.org/10.1090/bull/1528
Received by editor(s): October 2, 2015
Published electronically: April 6, 2016
Additional Notes: This article contains an extended exposition of the lectures given at Isaac Newton Institute for Mathematical Sciences in January 2015. The work was also partially supported by the NSF grant DMS # 1517938. The author expresses his gratitude to the INI and NSF for the support.
Dedicated: Dedicated to the memory of mathematicians and dear friends M. Birman, L. Ehrenpreis, S. Krein, V. Meshkov, and M. Novitskii
Article copyright: © Copyright 2016 American Mathematical Society