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Almost Complete Transmission of Waves Through Perforated Cross-Walls in a Waveguide with Dirichlet Boundary Condition

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Abstract

We consider a waveguide composed of two not necessity equal semi-infinite strips (trunks) and a rectangle (resonator) connected by narrow openings in the shared walls. As their diameter vanishes, we construct asymptotics for the scattering coefficients, justifying them by the technique of weighted spaces with detached asymptotics. We establish a criterion for the possibility of observing, at a given frequency, almost complete transmission of waves through both perforated cross-walls. This effect is revealed due to a fine-tuning of the resonator height and the criterion involves an equation relating some geometrical characteristics of the waveguide to the wave numbers in the trunks, while any mirror symmetry turns the criterion trivial.

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References

  1. Ladyzhenskaya O. A., The Boundary Value Problems of Mathematical Physics, Springer, New York etc. (1985).

    Book  Google Scholar 

  2. Birman M. Sh. and Solomjak M. Z., Spectral Theory of Self-Adjoint Operators in Hilbert Space, D. Reidel, Dordrecht (1987).

    Book  Google Scholar 

  3. Mittra R. and Lee S. W., Analytical Techniques in the Theory of Guided Waves, MacMillan, New York (1971).

    MATH  Google Scholar 

  4. Breuer H.-P. and Petruccione F., The Theory of Open Quantum Systems, Oxford Univ., Oxford (2002).

    MATH  Google Scholar 

  5. Schoelkopf R. J. and Girvin S. M., “Wiring up quantum systems,” Nature, vol. 51, 664–669 (2008).

    Article  Google Scholar 

  6. O’Brien J. L., Furusawa A., and Vuc̆ković J., “Photonic quantum technologies,” Nat. Photonics, vol. 3, 687–695 (2009).

    Article  Google Scholar 

  7. Exner P. and Kovarîk H., Quantum Waveguides. Theoretical and Mathematical Physics, Springer, Cham (2015).

    Book  Google Scholar 

  8. Weinstein L. A., The Theory of Diffraction and the Factorization Method, Golem, Boulder (1969).

    Google Scholar 

  9. Shanin A. V., “Weinstein’s diffraction problem: embedding formula and spectral equation in parabolic approximation,” SIAM J. Appl. Math., vol. 70, no. 4, 1201–1218 (2009).

    Article  MathSciNet  Google Scholar 

  10. Nazarov S. A., “Scattering anomalies in a resonator above the thresholds of the continuous spectrum,” Sb. Math., vol. 206, no. 6, 782–813 (2015).

    Article  MathSciNet  Google Scholar 

  11. Korolkov A. I., Nazarov S. A., and Shanin A. V., “Stabilizing solutions at thresholds of the continuous spectrum and anomalous transmission of waves,” Z. Angew. Math. Mech., vol. 96, no. 10, 1245–1260 (2016).

    Article  MathSciNet  Google Scholar 

  12. Shanin A. V. and Korolkov A. I., “Diffraction of a mode close to its cut-off by a transversal screen in a planar waveguide,” Wave Motion, vol. 68, 218–241 (2017).

    Article  MathSciNet  Google Scholar 

  13. Nazarov S. A., “Anomalies of acoustic wave scattering near the cut-off points of continuous spectrum (a review),” Acoust. Phys., vol. 66, no. 5, 477–494 (2020).

    Article  Google Scholar 

  14. Kriegsmann G. A., “Complete transmission through a two-dimensional diffraction grating,” SIAM J. Appl. Math., vol. 65, no. 1, 24–42 (2004).

    Article  MathSciNet  Google Scholar 

  15. Baskin L. M., Kabardov M., Neittaanmäki P., Plamenevskii B. A., and Sarafanov O. V., “Asymptotic and numerical study of resonant tunneling in two-dimensional quantum waveguides of variable cross section,” Comp. Math. Math. Phys., vol. 53, no. 11, 1664–1683 (2013).

    Article  MathSciNet  Google Scholar 

  16. Lin J. and Zhang H., “Scattering by a periodic array of subwavelength slits I: field enhancement in the diffraction regime,” Multiscale Model. Sim., vol. 16, no. 2, 922–953 (2018).

    Article  MathSciNet  Google Scholar 

  17. Delitsyn A. and Grebenkov D. S., “Mode matching methods for spectral and scattering problems,” Q. J. Mech. Appl. Math., vol. 71, no. 4, 537–580 (2018).

    MathSciNet  MATH  Google Scholar 

  18. Maz’ya V. G., Nazarov S. A., and Plamenevskii B. A., “On the asymptotic behavior of solutions of elliptic boundary value problems with irregular perturbations of the domain,” in: Problems of Mathematical Analysis [Russian]. Vol. 8, Leningrad Univ., Leningrad (1981), 72–153.

  19. Maz’ya V. G., Nazarov S. A., and Plamenevskii B. A., “Evaluation of the asymptotic form of the ‘intensity coefficients’ on approaching corner or conical points,” Comp. Math. Math. Phys., vol. 23, no. 2, 50–58 (1983).

    Article  Google Scholar 

  20. Maz’ya V. G., Nazarov S. A., and Plamenevskii B. A., “Asymptotic expansions of the eigenvalues of boundary value problems for the Laplace operator in domains with small holes,” Math. USSR-Izv., vol. 24, no. 2, 321–345 (1985).

    Article  Google Scholar 

  21. Maz’ya V., Nazarov S., and Plamenevskij B., Asymptotic Theory of Elliptic Boundary Value Problems in Singularly Perturbed Domains. Vol. 1 and 2, Birkhäuser, Basel (2000).

    Book  Google Scholar 

  22. Van Dyke M., Perturbation Methods in Fluid Mechanics, Academic, New York and London (1964).

    MATH  Google Scholar 

  23. Il’in A. M., Matching of Asymptotic Expansions of Solutions of Boundary Value Problems, Amer. Math. Soc., Providence (1992) (Transl. Math. Monogr., Vol. 102).

    Book  Google Scholar 

  24. Sedov L. I., A Course in Continuum Mechanics. Vol. II: Physical Functions and Formulations of Problems, Wolters-Noordhoff, Groningen (1972).

    MATH  Google Scholar 

  25. Kondrat’ev V. A., “Boundary problems for elliptic equations in domains with conical or angular points,” Trans. Moscow Math. Soc., vol. 16, 227–313 (1967).

    MathSciNet  MATH  Google Scholar 

  26. Nazarov S. A. and Plamenevsky B. A., Elliptic Problems in Domains with Piecewise Smooth Boundaries, De Gruyter, Berlin and New York (1994).

    Book  Google Scholar 

  27. Nazarov S. A., “The polynomial property of self-adjoint elliptic boundary-value problems and an algebraic description of their attributes,” Russian Math. Surveys, vol. 54, no. 5, 947–1014 (1999).

    Article  MathSciNet  Google Scholar 

  28. Vladimirov V. S., Generalized Functions in Mathematical Physics [Russian], Nauka, Moscow (1979).

    MATH  Google Scholar 

  29. Nazarov S. A. and Romashev Yu. A., “Variation of the intensity factor under rupture of the ligament between two collinear cracks,” Izv. Akad. Nauk Armenian SSR. Mekh., vol. 4, 30–40 (1982).

    MATH  Google Scholar 

  30. Vishik M. I. and Lyusternik L. A., “Regular degeneration and a boundary layer for linear differential equations with a small parameter,” Uspekhi Mat. Nauk, vol. 12, no. 5, 3–122 (1957).

    MathSciNet  MATH  Google Scholar 

  31. Nazarov S. A., “Variational and asymptotic methods for finding eigenvalues below the continuous spectrum threshold,” Sib. Math. J., vol. 51, no. 5, 866–878 (2010).

    Article  MathSciNet  Google Scholar 

  32. Nazarov S. A., “Asymptotic expansions of eigenvalues in the continuous spectrum of a regularly perturbed quantum waveguide,” Theoret. Math. Phys., vol. 167, no. 2, 606–627 (2011).

    Article  MathSciNet  Google Scholar 

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Funding

The work was supported by the Russian Science Foundation (Grant 17–11–01003).

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Authors and Affiliations

  1. St. Petersburg State University, St. Petersburg, Russia

    S. A. Nazarov

  2. INRIA/Centre de mathématiques appliquées, École Polytechnique, Palaiseau, France

    L. Chesnel

Authors
  1. S. A. Nazarov
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  2. L. Chesnel
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Correspondence to S. A. Nazarov.

Additional information

Translated from Sibirskii Matematicheskii Zhurnal, 2021, Vol. 62, No. 2, pp. 339–361. https://doi.org/10.33048/smzh.2021.62.208

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Nazarov, S.A., Chesnel, L. Almost Complete Transmission of Waves Through Perforated Cross-Walls in a Waveguide with Dirichlet Boundary Condition. Sib Math J 62, 272–291 (2021). https://doi.org/10.1134/S0037446621020087

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  • DOI: https://doi.org/10.1134/S0037446621020087

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