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Journal of Mathematical Sciences

, Volume 243, Issue 5, pp 746–773 | Cite as

Asymptotics of Eigenvalues in Spectral Gaps of Periodic Waveguides with Small Singular Perturbations

  • S. A. NazarovEmail author
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The asymptotics of eigenvalues appearing near the lower edge of a spectral gap of the Dirichlet problem is studied for the Laplace operator in a d-dimensional periodic waveguide with a singular perturbation of the boundary by creating a hole with a small diameter ε. Several versions of the structure of the gap edge are considered. As usual, the asymptotic formulas are different in the cases d ≥ 3 and d = 2, where the eigenvalues occur at distances O(ε2(d−2)) or O(ε2d) and O(|ln ε|−2) or O(ε4), respectively, from the gap edge. Other types of singular perturbation of the waveguide surface and other types of boundary conditions are discussed, which provide the appearance of eigenvalues near both edges of one or several gaps.

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References

  1. 1.
    O. A. Ladyzhenskaya , Boundary Value Problems of Mathematical Physics, Springer-Verlag, New York etc. (1985).zbMATHCrossRefGoogle Scholar
  2. 2.
    M. Sh. Birman and M. Z. Solomyak, Spectral Theory of Selfadjoint Operators in Hilbert Space [in Russian], Leningrad Univ., Leningrad (1980).Google Scholar
  3. 3.
    A. V. Sobolev and J. Walthoe, “Absolute continuity in periodic waveguides,” Proc. London Math. Soc., 85, No. 1, 717–741 (2002).MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    T. A. Suslina and R. G. Shterenberg, “Absolute continuity of the spectrum of the magnetic Schrödinger operator with a metric in a two-dimensional periodic waveguide,” Algebra Analiz, 14:2, 159–206 (2002).zbMATHGoogle Scholar
  5. 5.
    I. Kachkovskii and N. Filonov, “Absolute continuity of the spectrum of a periodic Schrödinger operator in a multidimensional cylinder,” Algebra Analiz, 21:1, 133–152 (2009).zbMATHGoogle Scholar
  6. 6.
    I. M. Gel’fand, “Expansion in eigenfunctions of an equation with periodic coefficients,” Dokl. Akad. Nauk SSSR, 73, 1117–1120 (1950).Google Scholar
  7. 7.
    S. A. Nazarov, “Elliptic boundary value problems with periodic coefficients in a cylinder,” Izv. Akad. Nauk SSSR, Ser. Mat., 45:1, 101–112 (1981).MathSciNetzbMATHGoogle Scholar
  8. 8.
    S. A. Nazarov and B. A. Plamenevskii, Elliptic Problems in Domains With Piecewise Smooth Boundaries [in Russian], Nauka, Moscow (1991).Google Scholar
  9. 9.
    T. Kato, Perturbation Theory for Linear Operators, Grundlehren Math. Wiss., 132, Springer-Verlag, New York (1966).Google Scholar
  10. 10.
    M. M. Skriganov, “Geometric and arithmetic methods in the spectral theory of multidimensional periodic operators,” Trudy Mat. Inst. Steklov., 171, 3–122 (1985).MathSciNetGoogle Scholar
  11. 11.
    P. Kuchment, Floquet Theory for Partial Differential Equations, Birchäuser, Basel (1993).zbMATHCrossRefGoogle Scholar
  12. 12.
    P. A. Kuchment, “Floquet theory for partial differential equations,” Uspekhi Mat. Nauk, 37:4 (226), 3–52 (1982).MathSciNetzbMATHGoogle Scholar
  13. 13.
    P. Kuchment, “The mathematics of photonic crystals,” Chap. 7 in Mathematical Modeling in Optical Science, in: Frontiers in Applied Mathematics, SIAM, 22 (2001), pp. 207–272.Google Scholar
  14. 14.
    S. A. Nazarov, “Properties of spectra of boundary value problems in cylindrical and quasicylindrical domain Sobolev Spaces in Mathematics,” Vol. II (Maz’ya V., Ed.) Intern. Math. Series, 9, 261–309 (2008).Google Scholar
  15. 15.
    W. Bulla, F. Gesztesy, W. Renger, and B. Simon, “Weakly coupled bound states in quantum waveguides,” Proc. Amer. Math. Soc., 125, No. 8, 1487–1495 (1997).MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    V. V. Grushin, “On the eigenvalues of finitely perturbed Laplace operators in infinite cylindrical domains,” Mat. Zamet., 75:3, 360–371 (2004).MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    R. R. Gadyl’shin, “Local perturbations of quantum waveguides,” Teoret. Mat. Fiz., 145:3, 358–371 (2005).MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    D. I. Borisov, “The discrete spectrum of a pair of nonsymmetric waveguides connected by a window,” Mat. Sborn., 197, No. 4, 3–32 (2006).CrossRefGoogle Scholar
  19. 19.
    S. A. Nazarov, “Variational and asymptotic methods for finding eigenvalues below the continuous spectrum threshold,” Sibirsk. Mat. Zh., 51, No. 5, 1086–1101 (2010).MathSciNetGoogle Scholar
  20. 20.
    M. Sh. Birman and M. Z. Solomyak, “Discrete negative spectrum under nonregular perturbations (polyharmonic operators, Schr¨odinger operators, with a magnetic fields, periodic operators),” in: Rigorous Results in Quantum Dynamics (Liblice, 1990), World Sci. Publishing, River Edge, NJ (1991), pp. 25–36.Google Scholar
  21. 21.
    M. Sh. Birman, ”The discrete spectrum of the periodic Schrödinger operator perturbed by a decreasing potential,” Algebra Analiz, 8, No. 1, 3–20 (1996).Google Scholar
  22. 22.
    M. Sh. Birman, “The discrete spectrum in gaps of the perturbed periodic Schrödinger operator. I. Regular perturbations,” in: Boundary Value Problems, Schrödinger Operators, Deformation Quantization. Math. Top., Akademie Verlag, Berlin, 8 (1995), pp. 334–352.Google Scholar
  23. 23.
    M. Sh. Birman, “The discrete spectrum in gaps of the perturbed periodic Schrödinger operator. II. Nonregular perturbations,” Algebra Analiz, 9, No. 6, 62–89 (1997).zbMATHGoogle Scholar
  24. 24.
    A. Figotin and A. Klein, “Midgap defect modes in dielectric and acoustic media,” SIAM J. Appl. Math., 58, No. 6, 1748–1773 (1998).MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    H. Ammari and F. Santosa, “Guided waves in a photonic bandgap structure with a line defect,” SIAM J. Appl. Math., 64, No. 6, 2018–2033 (2004).MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    D. Miao and F. Ma, “On guided waves created by line defects,” J. Stat. Phys., 130, 1197–1215 (2008).MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    S. A. Nazarov, “Gaps and eigenfrequencies in the spectrum of a periodic acoustic waveguide,” Akustik Zh., 59, No. 3, 312–321 (2013).Google Scholar
  28. 28.
    B. M. Brown, V. Hoang, M. Plum, and I. Wood, “Spectrum created by line defects in periodic structures,” Math. Nachr., 287, 1972–1985 (2014).MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    S. A. Nazarov, “Bounded solutions in a T-shaped waveguide and the spectral properties of the Dirichlet ladder,” Zh. Vychisl. Mat. Mat. Fiz., 54, No. 8, 1299–1318 (2014).MathSciNetzbMATHGoogle Scholar
  30. 30.
    B. Delourme, S. Fliss, P. Joly, and E. Vasilevskaya, “Trapped modes in thin and infinite ladder like domains. Part 1: Existence results,” Asymptotic Analysis, 103, No. 3, 103–134 (2017).MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    S. A. Nazarov, “Asymptotics of eigenvalues in spectral gaps under regular perturbations of walls of a periodic waveguide,” Probl. Mat. Analiz., Novosibirsk, Vol. 89, 63–98 (2017).Google Scholar
  32. 32.
    S. A. Nazarov, “Almost standing waves in a periodic waveguide with resonator, and nearthreshold eigenvalues,” Algebra Analiz, 28, No. 3, 111–160 (2016).Google Scholar
  33. 33.
    D. V. Evans, M. Levitin, and D. Vasil’ev, “Existence theorems for trapped modes,” J. Fluid Mech., 261, 21–31 (1994).MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    S. A. Nazarov, “Asymptotic expansions of eigenvalues in the continuous spectrum of a regularly perturbed quantum waveguide,” Theoret. Mat. Fiz., 167, No. 2, 239–262 (2011).MathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    S. A. Nazarov, “Enforced stability of a simple eigenvalue in the continuous spectrum,” Funkt. Anal. Prilozhen., 47, No. 3, 37–53 (2013).CrossRefGoogle Scholar
  36. 36.
    I. C. Gokhberg and M. G. Kreyn, Introduction to the Theory of Linear not Self-Adjoint operators, Nauka, Moscow (1965).Google Scholar
  37. 37.
    M. M. Vainberg and V. A. Trenogin, Theory of Branching of Solutions of Non-Linear Equations [in Russian], Nauka, Moscow (1969).zbMATHGoogle Scholar
  38. 38.
    W. G. Mazja, S. A. Nazarov, and B. A. Plamenewski, Asymptotische Theorie Elliptischer Randwertaufgaben in Singulär Gestörten Gebieten, Vol. 1, Akademie-Verlag, Berlin (1991).Google Scholar
  39. 39.
    M. Van Dyke, Perturbation Methods in Fluid Mechanics [Russian translation], Mir, Moscow (1967).zbMATHGoogle Scholar
  40. 40.
    A. M. Il’in, Matching of Asymptotic Expansions of Solutions of Boundary-Value Problems [in Russian], Nauka, Moscow (1989).Google Scholar
  41. 41.
    S. A. Nazarov, “Opening a gap in the continuous spectrum of a periodically perturbed waveguide,” Mat. Zamet., 87, No. 5, 764–786 (2010).MathSciNetCrossRefGoogle Scholar
  42. 42.
    F. L. Bakharev, S. A. Nazarov, and K. M. Ruotsalainen, “A gap in the spectrum of the Neumann–Laplacian on a periodic waveguide,” Appl. Analys., 88, 1–17 (2012).zbMATHGoogle Scholar
  43. 43.
    D. Borisov and K. Pankrashkin, “Quantum waveguides with small periodic perturbations: gaps and edges of Brillouin zones,” J. Physics A: Math. Theor., 46, No. 23, 203–235 (2013).MathSciNetzbMATHCrossRefGoogle Scholar
  44. 44.
    S. A. Nazarov, “Asymptotic behavior of spectral gaps in a regularly perturbed periodic waveguide,” Vestn. St.-Petersb. Univ., Ser. 1,2, No. 7, 54–63 (2013).Google Scholar
  45. 45.
    V. G. Maz’ya, S. A. Nazarov, and B. A. Plamenevskii, “Asymptotic expansions of eigenvalues of boundary value problems for the Laplace operator in domains with small holes,” Izv. Akad. Nauk SSSR, Ser. Mat., 48, No. 2, 347–371 (1984).zbMATHCrossRefGoogle Scholar
  46. 46.
    G. Polya and G. Szegö, Isoperimetric Inequalities in Mathematical Physics [Russian translation], Fizmatgiz, Moscow (1962).zbMATHGoogle Scholar
  47. 47.
    N. S. Landkof, Foundations of Modern Potential Theory [in Russian], Nauka, Moscow (1966).Google Scholar
  48. 48.
    V. A. Kondrat’ev, “Boundary-value problems for elliptic equations in domains with conical or corner points,” Trudy Moskov. Mat. Obshch., 16, 209–292 (1967).Google Scholar
  49. 49.
    S. A. Nazarov, “On the constants in the asymptotic expansion of solutions of elliptic boundary value problems with periodic coefficients in a cylinder,” Vestn. Leningr. Univ., Ser. 1,3, No. 15, 16–22 (1985).Google Scholar
  50. 50.
    S. A. Nazarov, “The asymptotics of frequencies of elastic waves trapped by a small crack in an anisotropic waveguide,” Mekh. tverd. tela, No. 6, 112–122 (2010).Google Scholar
  51. 51.
    S. A. Nazarov, M. Specovius-Neugebauer, and J. Sokolowski, “Polarization matrices in anisotropic heterogeneous elasticity,” Asymp. Analys., 68, No. 4, 189–249 (2010).MathSciNetzbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.St. Petersburg State UniversitySt. PetersburgRussia

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