Advertisement

Journal of Mathematical Sciences

, Volume 210, Issue 5, pp 590–621 | Cite as

Perturbation of Threshold of Essential Spectrum for Waveguides with Windows. II: Asymptotics

  • D. I. Borisov
Article

We consider a quantum waveguide described by a pair of three-dimensional plane-parallel layers with common boundary containing a window joining the layers. As the window expands, the threshold of the essential spectrum generates bound states, anti-bound states, or resonances. We propose an original efficient algorithm for constructing complete asymptotic expansions for the associated spectral parameters and nontrivial solutions. Using a nontrivial technique, we justify the constructed asymptotic expansions and analyze their structure. Bibliography: 36 titles. Illustrations: 1 figure.

Keywords

Asymptotic Expansion Holomorphic Function Small Neighborhood Nontrivial Solution Resonance State 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    D. I. Borisov, “Perturbation of threshold of essential spectrum for waveguides with windows. I: Decreasing resonance solutions” [in Russian], Probl. Mat. Anal. 77, 19–54 (2014); English transl.: J. Math. Sci., New York 205, No. 2, 141–181 (2015).Google Scholar
  2. 2.
    D. Borisov, “The spectrum of two quantum layers coupled by a window,” J. Phys. A. Math. Theor. 40, No. 19, 5045–5066 (2007).zbMATHCrossRefGoogle Scholar
  3. 3.
    D. I. Borisov, “Discrete spectrum of an asymmetric pair of waveguides coupled through a window” [in Russian], Mat. Sb. 197, No. 4, 3–32 (2006); English trans.: Sb. Math. 197, No. 4, 475-504 (2006).Google Scholar
  4. 4.
    S. A. Nazarov, “Asymptotics of an eigenvalue on the continuous spectrum of two quantum waveguides coupled through narrow windows” [in Russian], Mat. Zametki 93, No. 2, 227–245 (2013); English transl.: Math. Notes 93, No. 2, 266–281 (2013).Google Scholar
  5. 5.
    D. Borisov, P. Exner, and R. Gadyl’shin, “Geometric coupling thresholds in a twodimensional strip,” J. Math. Phys. 43, No. 12, 6265–6278 (2002).zbMATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    D. Borisov, T. Ekholm, and H. Kovařík, “Spectrum of the magnetic Schrödinger operator in a waveguide with combined boundary conditions,” Ann. Henri Poincaré. 6, No. 2, 327–342 (2005).zbMATHCrossRefGoogle Scholar
  7. 7.
    D. Borisov and P. Exner, “Exponential splitting of bound states in a waveguide with a pair of distant windows,” J. Phys. A: Math. Gen. 37, No. 10, 3411–3428 (2004).zbMATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    D. Borisov and P. Exner, “Distant perturbation asymptotics in window-coupled waveguides. I. The non-threshold case,” J. Math. Phys. 47, No. 11, 113502-1–113502-24 (2006).Google Scholar
  9. 9.
    W. Bulla, F. Gesztesy, W. Renger, and B. Simon, “Weakly coupled bound states in quantum waveguides,” Proc. Am. Math. Soc. 125, No. 5, 1487–1495 (1997).zbMATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    P. Exner, P. Šeba, M. Tater, and D. Vaněck, “Bound states and scattering in quantum waveguides coupled laterally through a boundary window,” J. Math. Phys. 37, No. 10, 4867–4887 (1996).zbMATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    P. Exner and S. Vugalter, “Asymptotics estimates for bound states in quantum waveguides coupled laterally through a narrow window,” Ann. Inst. H. Poincaré Phys. Théor. 65, No. 1, 109-123 (1996).zbMATHMathSciNetGoogle Scholar
  12. 12.
    P. Exner and S. Vugalter, “Bound-state asymptotic estimate for window-coupled Dirichlet strips and layers,” J. Phys. A. Math. Gen. 30, No. 22, 7863–7878 (1997).zbMATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    R. R. Gadyl’shin, “On regular and singular perturbations of acoustic and quantum waveguides,” C.R. Mech. 332, No. 8, 647–652 (2004).Google Scholar
  14. 14.
    Y. Hirayama, Y. Tokura, A. D. Wieck, S. Koch, R. J. Haug, K. von Klitzing, and K. Ploog, “Transport characteristics of a window-coupled in-plane-gated wire system,” Phys. Rev. B. 48, No. 11, 7991–7998 (1993).CrossRefGoogle Scholar
  15. 15.
    Ch. Kunze, “Leaky and mutually coupled wires,” Phys. Rev. B 48, No. 19, 14338–14346 (1993).CrossRefGoogle Scholar
  16. 16.
    O. Olendski and L. Mikhailovska, “A straight quantum wave guide with mixed Dirichlet and Neumann boundary conditions in uniform magnetic fields,” J. Phys. A: Math. Theor. 40, No. 17, 4609-4634 (2007).zbMATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    I. Yu. Popov, “Asymptotics of bound states and bands for laterally coupled waveguides and layers,” J. Math. Phys. 43, No. 1, 215–234 (2002).zbMATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    H. Najar and O. Olendski, “Spectral and localization properties of the Dirichlet wave guide with two concentric Neumann discs,” J. Phys. A: Math. Theor. 44, No. 30, id 305304 (2011).Google Scholar
  19. 19.
    H. Najar, S. B. Hariz, and M. B. Salah, “On the discrete spectrum of a spatial quantum waveguide with a disc window,” Math. Phys. Anal. Geom. 13, No. 1, 19–28 (2010).zbMATHMathSciNetCrossRefGoogle Scholar
  20. 20.
    H. Najar and M. Raissi, “A quantum waveguide with Aharonov–Bohm magnetic field,” Math. Meth. Appl. Sci. online first, DOI:  10.1002/mma.3462 (2015).
  21. 21.
    D. Borisov and G. Cardone, “Homogenization of the planar waveguide with frequently alternating boundary conditions,” J. Phys. A. Math. Gen. 42, No. 36, id 365205 (2009).Google Scholar
  22. 22.
    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).zbMATHMathSciNetCrossRefGoogle Scholar
  23. 23.
    D. Borisov, R. Bunoiu, and G. Cardone, “Homogenization and asymptotics for a waveguide with an infinite number of closely located small windows” [in Russian], Probl. Mat. Anal. 58, 59–68 (2011); English transl.: J. Math. Sci., New York 176, No. 6, 774–785 (2011).Google Scholar
  24. 24.
    D. Borisov and R. Bunoiu, G. Cardone, “On a waveguide with an infinite number of small windows,” C. R. Math. 349, No. 1-2, 53–56 (2011).Google Scholar
  25. 25.
    D. Borisov, R. Bunoiu, and G. Cardone, “Waveguide with non-periodically alternating Dirichlet and Robin conditions: homogenization and asymptotics,” Z. Angew. Math. Phys. 64, No. 3, 439–472 (2013).zbMATHMathSciNetCrossRefGoogle Scholar
  26. 26.
    D. I. Borisov and K. V. Pankrashkin, “Gap opening and split band edges in waveguides coupled by a periodic system of small windows” [in Russian], Mat. Zametki 93, No. 5, 665–683 (2013); English transl.: Math. Notes 93, No. 5, 660–675 (2013).Google Scholar
  27. 27.
    D. I. Borisov and K. V. Pankrashkin, “On the extrema of band functions in periodic waveguides” [in Russian], Funkts. Anal. Prilozh. 47, No. 3, 87–90 (2013); English transl.: Funct. Anal. Appl. 47, No. 3, 238–240 (2013).Google Scholar
  28. 28.
    F. Gesztesy and H. Holden, “A unified approach to eigenvalues and resonances of Schrödinger operators using Fredholm determinants,” J. Math. Anal. Appl. 123, No. 1, 181–198 (1987).zbMATHMathSciNetCrossRefGoogle Scholar
  29. 29.
    B. Simon, “The bound state of weakly coupled Schrödinger operators in one and two dimensions,” Ann. Phys. 97, No. 2, 279–288 (1976).zbMATHCrossRefGoogle Scholar
  30. 30.
    M. Klaus and B. Simon, “Coupling constant thresholds in nonrelativistic quantum mechanics. I. Short-range two-body case,” Ann. Phys. 130, No. 2, 251–281 (1980).zbMATHMathSciNetCrossRefGoogle Scholar
  31. 31.
    M. Klaus, “On the bound state of Schrödinger operators in one dimension,” Ann. Phys. 108, No. 2, 288–300 (1977).zbMATHMathSciNetCrossRefGoogle Scholar
  32. 32.
    D. Borisov and G. Cardone, “Planar waveguide with “twisted” boundary conditions: discrete spectrum,” J. Math. Phys. 52, No. 12, id 123513 (2011).Google Scholar
  33. 33.
    S. A. Nazarov, “Artificial boundary conditions for finding surface waves in the problem of diffraction by a periodic boundary” [in Russian], Zh. Vych. Mat. Mat. Phys. 46, No. 12, 2265–2276 (2006); English transl.: Comput. Math. Math. Phys. 46, No. 12, 2164–2175 (2006).Google Scholar
  34. 34.
    A. M. Il’in, Matching of Asymptotic Expansions of Solutions of Boundary Value Problems [in Russian], Nauka, Moscow (1989); English transl.: Am. Math. Soc., Providence, RI (1992).Google Scholar
  35. 35.
    S. A. Nazarov, “Asymptotics of eigenfrequencies in the spectral gaps caused by a perturbation of a periodic waveguide” [in Russian], Dokl. Akad. Nauk, Ross. Akad. Nauk 447, No. 4, 382–386 (2012); English transl.: Dokl. Math. 86, No. 3, 871–875 (2012).Google Scholar
  36. 36.
    G. N.Watson, A Treatise of the Theory of Bessel Functions, Cambridge Univ. Press, London (1966).Google Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Institute of MathematicsUSC RASUfaRussia
  2. 2.Bashkir State Pedagogical UniversityUfaRussia
  3. 3.University of Hradec KrálovéHradec KrálovéCzech Republic

Personalised recommendations