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Russian Mathematical Surveys, 2006, Volume 61, Issue 4, Pages 663–766
DOI: https://doi.org/10.1070/RM2006v061n04ABEH004343
(Mi rm2121)
 

This article is cited in 129 scientific papers (total in 129 papers)

Instability zones of periodic 1-dimensional Schrödinger and Dirac operators

P. B. Djakova, B. S. Mityaginb

a Sofia University St. Kliment Ohridski, Faculty of Mathematics and Computer Science
b Ohio State University
References:
Abstract: The spectra of Schrödinger and Dirac operators with periodic potentials on the real line $\mathbb R$ have a band structure, that is, the intervals of continuous spectrum alternate with spectral gaps, or instability zones. The sizes of these zones decay, and the rate of decay depends on the smoothness of the potential. In the opposite direction, one can make conclusions about the smoothness of a potential based on the rate of decay of the instability zones. In the 1960s and 1970s this phenomenon was understood at the level of infinitely differentiable or analytic functions in the case of Schrödinger operators. However, only recently has the relationship between the smoothness of the potential and the rate of decay of the instability zones become completely understood and analyzed
  • for a broad range of classes of differentiable functions,
  • for Dirac operators and not just for Hill–Schrödinger operators,
  • in both the self-adjoint and non-self-adjoint cases.
This paper is devoted to a survey of these results, mostly with complete proofs based on an approach developed by the authors.
Received: 23.04.2006
Russian version:
Uspekhi Matematicheskikh Nauk, 2006, Volume 61, Issue 4(370), Pages 77–182
DOI: https://doi.org/10.4213/rm2121
Bibliographic databases:
Document Type: Article
UDC: 517.927+517.984
MSC: Primary 47E05, 34L40, 34L20; Secondary 34B05, 34L10
Language: English
Original paper language: Russian
Citation: P. B. Djakov, B. S. Mityagin, “Instability zones of periodic 1-dimensional Schrödinger and Dirac operators”, Uspekhi Mat. Nauk, 61:4(370) (2006), 77–182; Russian Math. Surveys, 61:4 (2006), 663–766
Citation in format AMSBIB
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\paper Instability zones of periodic 1-dimensional Schr\"odinger and Dirac operators
\jour Uspekhi Mat. Nauk
\yr 2006
\vol 61
\issue 4(370)
\pages 77--182
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\pages 663--766
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  • https://www.mathnet.ru/eng/rm2121
  • https://doi.org/10.1070/RM2006v061n04ABEH004343
  • https://www.mathnet.ru/eng/rm/v61/i4/p77
  • This publication is cited in the following 129 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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    Успехи математических наук Russian Mathematical Surveys
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    Abstract page:1847
    Russian version PDF:680
    English version PDF:30
    References:124
    First page:11
     
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