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Homogenization of the two-dimensional periodic Dirac operator

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We consider the homogenization problem for the two-dimensional periodic Dirac operator with magnetic potential of two types: singular and nonsingular. Based on the operator-theoretic approach due to M. Birman and T. Suslina, we approximate the resolvent in the space of bounded operators acting in \( {L_2}\left( {{{\mathbb{R}}^2};{{\mathbb{C}}^2}} \right) \) with order-sharp error estimates. Bibliography: 4 titles.

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References

  1. M. Sh. Birman and T. A. Suslina, “Second order periodic differential operators. Threshold properties and homogenization” [in Russian], Algebra Anal. 15, No. 5, 1–108 (2003); English transl.: St. Petersbg. Math. J. 15, No. 5, 639–714 (2004).

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  2. M. Sh. Birman and T. A. Suslina, “Homogenization with corrector term for periodic elliptic differential operators” [in Russian], Algebra Anal. 17, No. 6, 1–104 (2005). English transl.: St. Petersbg. Math. J. 17, No. 6, 897–973 (2006).

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  3. M. Sh. Birman and T. A. Suslina, “Homogenization with corrector for periodic differential operators. Approximation of solutions in the Sobolev class \( {H^1}\left( {{{\mathbb{R}}^d}} \right) \)” [in Russian], Algebra Anal. 18, No. 6, 1–130 (2006); English transl.: St. Petersbg. Math. J. 18, No. 6, 857–955 (2007).

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  4. T. A. Suslina, “Homogenization in the Sobolev class \( {H^1}\left( {{{\mathbb{R}}^d}} \right) \) for second-order periodic elliptic operators with the inclusion of first-order terms” [in Russian], Algebra Anal. 22, No. 1, 108–222 (2010); English transl.: St. Petersbg. Math. J. 22, No. 1, 81–162 (2011).

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  1. St. Petersburg State University, Institute of Physics Ulyanovskaya 1, Petrodvorets, St. Petersburg, 198504, Russia

    A. A. Kukushkin

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  1. A. A. Kukushkin
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Correspondence to A. A. Kukushkin.

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Translated from Problemy Matematicheskogo Analiza 68, January 2013, pp. 141–156.

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Kukushkin, A.A. Homogenization of the two-dimensional periodic Dirac operator. J Math Sci 189, 490–507 (2013). https://doi.org/10.1007/s10958-013-1202-3

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  • DOI: https://doi.org/10.1007/s10958-013-1202-3

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