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On Lacunas in the Lower Part of the Spectrum of the Periodic Magnetic Operator in a Strip

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Abstract

We consider the Schrödinger periodic magnetic operator in an infinite flat straight strip. We prove that if the magnetic potential satisfies certain conditions and the period is small enough, then the lower part of the band spectrum has no inner lacunas. The length of the lower part of the band spectrum with no inner lacunas is calculated explicitly. The upper estimate for the small parameter allowing these results is calculated as a number as well.

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References

  1. G. Barbatis and L. Parnovski, “Bethe–Sommerfeld conjecture for pseudo-differential perturbation,” Commun. Part. Differ. Equ., 34, No. 4, 383–418 (2009).

    Article  Google Scholar 

  2. D. I. Borisov, “On absence of lacunas in the lower part of Laplacian spectrum with fast alternation of boundary conditions in a strip,” Teor. Mat. Fiz., to be published.

  3. D. Borisov, R. Bunoiu, and G. Cardone, “On a waveguide with frequently alternating boundary conditions: homogenized Neumann condition,” Ann. Henri Poincaré, 11, No. 8, 1591–1627 (2010).

    Article  MathSciNet  Google Scholar 

  4. D. Borisov, R. Bunoiu, and G. Cardone, “Waveguide with nonperiodically alternating Dirichlet and Robin conditions: homogenization and asymptotics,” Z. Angew. Math. Phys., 64, No. 3, 439–472 (2013).

    Article  MathSciNet  Google Scholar 

  5. D. Borisov and G. Cardone, “Homogenization of the planar waveguide with frequently alternating boundary conditions,” J. Phys. A. Math. Gen., 42, No. 36, 365205 (2009).

    Article  MathSciNet  Google Scholar 

  6. D. Borisov, G. Cardone, and T. Durante, “Homogenization and uniform resolvent convergence for elliptic operators in a strip perforated along a curve,” Proc. R. Soc. Edin. Sec. A. Math., 146, No. 6, 1115–1158 (2016).

    Article  Google Scholar 

  7. D. Borisov, G. Cardone, L. Faella, and C. Perugia, “Uniform resolvent convergence for a strip with fast oscillating boundary,” J. Differ. Equ., 255, No. 12, 4378–4402 (2013).

    Article  MathSciNet  Google Scholar 

  8. B. E. Dahlberg and E. Trubowitz, “A remark on two dimensional periodic potentials,” Comment. Math. Helv., 57, No. 1, 130–134 (1982).

    Article  MathSciNet  Google Scholar 

  9. B. Helffer and A. Mohamed, “Asymptotics of the density of states for the Schrödinger operator with periodic electric potential,” Duke Math. J., 92, No. 1, 1–60 (1998).

    Article  MathSciNet  Google Scholar 

  10. Y. Karpeshina, “Spectral properties of the periodic magnetic Schrödinger operator in the highenergy region. Two-dimensional case,” Commun. Math. Phys., 251, No. 3, 473–514 (2004).

    Article  MathSciNet  Google Scholar 

  11. E. Lieb and M. Loss, Analysis [Russian translation], Nauchnaya Kniga, Novosibirsk (1998).

    MATH  Google Scholar 

  12. A. Mohamed, “Asymptotic of the density of states for the Schr¨odinger operator with periodic electromagnetic potential,” J. Math. Phys., 38, No. 8, 4023–4051 (1997).

    Article  MathSciNet  Google Scholar 

  13. L. Parnovski, “Bethe–Sommerfeld conjecture,” Ann. Henri Poincaré, 9, No. 3, 457–508 (2008).

    Article  MathSciNet  Google Scholar 

  14. L. Parnovski and A. Sobolev, “On the Bethe–Sommerfeld conjecture for the polyharmonic operator,” Duke Math. J., 107, No. 2, 209–238 (2001).

    Article  MathSciNet  Google Scholar 

  15. L. Parnovski and A. V. Sobolev, “Bethe–Sommerfeld conjecture for periodic operators with strong perturbations,” Invent. Math., 181, No. 3, 467–540 (2010).

    Article  MathSciNet  Google Scholar 

  16. N. N. Senik, “Averaging of a periodic elliptic operator in a strip under various boundary conditions,” Algebra i Analiz, 25, No. 4, 182–259 (2013).

    Google Scholar 

  17. M. M. Skriganov, “Geometric and arithmetic methods in the spectral theory of multidimensional periodic operators,” Tr. MIAN, 171, 3–122 (1985).

    MathSciNet  Google Scholar 

  18. M. M. Skriganov and A. V. Sobolev, “Asymptotic estimates for spectral zones of periodic Schr¨odinger operators,” Algebra i Analiz, 17, No. 1, 276–288 (2005).

    Google Scholar 

  19. M. M. Skriganov and A. V. Sobolev, “Variation of the number of lattice points in large balls,” Acta Arith., 120, No. 3, 245–267 (2005).

    Article  MathSciNet  Google Scholar 

  20. T. A. Suslina, “On averaging of periodic elliptic operator in a strip,” Algebra i Analiz, 16, No. 1, 269–292 (2004).

    MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Institute of Mathematics with Computer Center of the Ufa Science Center of the Russian Academy of Sciences, Ufa, Russia

    D. I. Borisov

  2. Bashkir State Pedagogical University, Ufa, Russia

    D. I. Borisov

  3. University of Hradec Králové, Hradec Králové, Czech Republic

    D. I. Borisov

Authors
  1. D. I. Borisov
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Correspondence to D. I. Borisov.

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Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 63, No. 3, Differential and Functional Differential Equations, 2017.

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Borisov, D.I. On Lacunas in the Lower Part of the Spectrum of the Periodic Magnetic Operator in a Strip. J Math Sci 253, 599–617 (2021). https://doi.org/10.1007/s10958-021-05257-x

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  • DOI: https://doi.org/10.1007/s10958-021-05257-x

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