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TMF, 2000, Volume 124, Number 1, Pages 3–17 (Mi tmf622)  

This article is cited in 15 scientific papers (total in 15 papers)

Spectrum of the periodic Dirac operator

L. I. Danilov

Physical-Technical Institute of the Ural Branch of the Russian Academy of Sciences

Abstract: The absolute continuity of the spectrum for the periodic Dirac operator
$$ \widehat D=\sum_{j=1}^n(-i\frac{\partial}{{\partial}x_j}-A_j) \widehat\alpha_j+\widehat V^{(0)}+\widehat V^{(1)},\quad x\in\mathbb R^n,\quad n\geq3, $$
is proved given that $A\in C(\mathbb R^n;\mathbb R^n)\cap H_\mathrm{loc}^q(\mathbb R^n;\mathbb R^n)$, $2q>n-2$, and also that the Fourier series of the vector potential $A\colon\mathbb R^n\to\mathbb R^n$ is absolutely convergent. Here, $\widehat V^{(s)}=(\widehat V^{(s)})^*$ are continuous matrix functions and $\widehat V^{(s)}\widehat\alpha_j=(-1)^s\widehat\alpha_j\widehat V^{(s)}$ for all anticommuting Hermitian matrices $\widehat\alpha_j$, $\widehat\alpha_j^2=\hat I$, $s=0,1$.

DOI: https://doi.org/10.4213/tmf622

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English version:
Theoretical and Mathematical Physics, 2000, 124:1, 859–871

Bibliographic databases:

Received: 29.06.1999
Revised: 27.10.1999

Citation: L. I. Danilov, “Spectrum of the periodic Dirac operator”, TMF, 124:1 (2000), 3–17; Theoret. and Math. Phys., 124:1 (2000), 859–871

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Kuchment, P, “On the structure of spectra of periodic elliptic operators”, Transactions of the American Mathematical Society, 354:2 (2001), 537  crossref  mathscinet  isi  scopus  scopus
    2. L. I. Danilov, “O spektre dvumernykh periodicheskikh operatorov Shredingera i Diraka”, Izv. IMI UdGU, 2002, no. 3(26), 3–98  mathnet
    3. V. M. Zhuravlev, “Autowaves in double-wire lines with the exponential-type nonlinear active element”, JETP Letters, 75:1 (2002), 9–14  mathnet  crossref
    4. L. I. Danilov, “Absolute Continuity of the Spectrum of a Periodic Schrödinger Operator”, Math. Notes, 73:1 (2003), 46–57  mathnet  crossref  crossref  mathscinet  zmath  isi
    5. L. I. Danilov, “The Spectrum of the Two-Dimensional Periodic Schrödinger Operator”, Theoret. and Math. Phys., 134:3 (2003), 392–403  mathnet  crossref  crossref  mathscinet  zmath  isi
    6. L. I. Danilov, “The absence of eigenvalues in the spectrum of ageneralized two-dimensional periodic Dirac operator”, St. Petersburg Math. J., 17:3 (2006), 409–433  mathnet  crossref  mathscinet  zmath
    7. L. I. Danilov, “Ob absolyutnoi nepreryvnosti spektra trekhmernogo periodicheskogo operatora Diraka”, Izv. IMI UdGU, 2006, no. 1(35), 49–76  mathnet
    8. Shen, ZW, “Uniform Sobolev inequalities and absolute continuity of periodic operators”, Transactions of the American Mathematical Society, 360:4 (2008), 1741  crossref  mathscinet  zmath  isi  scopus  scopus
    9. L. I. Danilov, “Absolyutnaya nepreryvnost spektra mnogomernogo periodicheskogo magnitnogo operatora Diraka”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 2008, no. 1, 61–96  mathnet
    10. Danilov, LI, “On absolute continuity of the spectrum of a periodic magnetic Schrodinger operator”, Journal of Physics A-Mathematical and Theoretical, 42:27 (2009), 275204  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    11. Danilov L.I., “On absolute continuity of the spectrum of three- and four-dimensional periodic Schrodinger operators”, J. Phys. A: Math. Theor., 43:21 (2010), 215201  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    12. Danilov L.I., “On Absolute Continuity of the Spectrum of a 3D Periodic Magnetic Dirac Operator”, Integral Equations Operator Theory, 71:4 (2011), 535–556  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    13. L. I. Danilov, “O spektre periodicheskogo operatora Shredingera s potentsialom iz prostranstva Morri”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 2012, no. 3, 25–47  mathnet
    14. L. I. Danilov, “O spektre periodicheskogo magnitnogo operatora Diraka”, Izv. IMI UdGU, 2016, no. 2(48), 3–21  mathnet  elib
    15. Kuchment P., “An overview of periodic elliptic operators”, Bull. Amer. Math. Soc., 53:3 (2016), 343–414  crossref  mathscinet  zmath  isi  elib  scopus
  • Теоретическая и математическая физика Theoretical and Mathematical Physics
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