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On local perturbations of waveguides

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Abstract

The paper deals with an arbitrary sufficiently small localized perturbation of waveguides with different types of boundary conditions. We study both the qualitative structure of the spectrum of the perturbed operator and conditions for the occurrence of eigenvalues from the continuous spectrum. For the case in which eigenvalues occur, their asymptotic behavior is obtained.

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References

  1. P. Exner and P. Seba, “Bound States in Curved Quantum Waveguides,” J. Math. Phys. 30, 2574–2580 (1989).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  2. P. Exner, “Bound States in Quantum Waveguides of a Slowly Decaying Curvature,” J. Math. Phys. 34, 23–28 (1993).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  3. P. Duclos and P. Exner, “Curvature–Induced Bound States in Quantum Waveguides in Two and Three Dimensions,” Rev. Math. Phys. 7, 73–102 (1995).

    Article  MathSciNet  MATH  Google Scholar 

  4. W. Bulla, F. Gesztesy, W. Renger and B. Simon, “Weakly Coupled Bound States in Quantum Waveguides,” Proc. Amer. Math. Soc. 127, 1487–1495 (1997).

    Article  MathSciNet  MATH  Google Scholar 

  5. P. Exner and S. A. Vugalter, “Bound States in a Locally Deformed Waveguide: the Critical Case,” Lett. Math. Phys. 39, 59–68 (1997).

    Article  MathSciNet  MATH  Google Scholar 

  6. D. Borisov, P. Exner, R. Gadyl’shin, and D. Krejcirik, “Bound States in Weakly Deformed Strips and Layers,” Ann. Henri Poincaré 2, 553–572 (2001).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  7. V. V. Grushin, “On the Eigenvalues of Finitely Perturbed Laplace Operators in Infinite Cylindrical Domains,” Mat. Zametki 75 (3), 360–371 (2004) [Math. Notes 75 (3), 331–340 (2004)].

    Article  MathSciNet  MATH  Google Scholar 

  8. V. V. Grushin, “Asymptotic Behavior of the Eigenvalues of the Schrodinger Operator with Transversal Potential in a Weakly Curved Infinite Cylinder,” Mat. Zametki 77 (5), 656–664 (2005) [Math. Notes 77 (5), 606–613 (2005)].

    Article  MathSciNet  MATH  Google Scholar 

  9. V. V. Grushin, “Asymptotic Behavior of Eigenvalues of the Laplace Operator in Infinite Cylinders Perturbed by Transverse Extensions,” Mat. Zametki 81 (3), 328–334 (2007) [Math. Notes 81 (3), 291–296 (2007)].

    Article  MathSciNet  MATH  Google Scholar 

  10. R. R. Gadyl’shin, “On Regular and Singular Perturbations of Acoustic and Quantum Waveguides,” Acad. Sci. Paris Ser. II 332 (8), 647–652 (2004).

    MATH  Google Scholar 

  11. R. R. Gadyl’shin, “Local perturbations of Quantum Waveguides,” Theoret. and Math. Phys. 145 (3), 358–371 (2005) [Teoret. Mat. Fiz. 145 (3), 358–371 (2005)].

    MathSciNet  MATH  Google Scholar 

  12. A. Bikmetov and R. Gadyl’shin, “On Quantum Waveguide with Shrinking Potential,” Russ. J. Math. Phys. 17 (1), 19–25 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  13. S. A. Nazarov, “Asymptotic Expansions of Eigenvalues in the Continuous Spectrum of a Regularly Perturbed Quantum Waveguide,” Teoret. and Mat. Fiz. 167 (2), 239–263 (2011) [Theoret. and Math. Phys. 167 (2), 606–627 (2011)].

    Article  MathSciNet  MATH  Google Scholar 

  14. G. Cardone, S. A. Nazarov, and K. Ruotsalainen, “Asymptotic Behaviour of an Eigenvalue in the Continuous Spectrum of a Narrowed Waveguide,” Mat. Sb. 203 (2), 3–32 (2012) [Sb. Math. 203 (2), 153–182 (2012)].

    Article  MathSciNet  MATH  Google Scholar 

  15. S. A. Nazarov, “Enforced Stability of an Eigenvalue in the Continuous Spectrum of a Waveguide with an Obstacle,” Zh. Vychisl. Mat. Mat. Fiz. 52 (3), 521–538 (2012) [Comput. Math. Math. Phys. 52 (3), 448–464 (2012)].

    MathSciNet  MATH  Google Scholar 

  16. S. A. Nazarov, “Scheme for Interpretation of Approximately Computed Eigenvalues Embedded in a Continuous Spectrum,” Zh. Vychisl. Mat. Mat. Fiz. 53 (6), 878–897 (2013) [Comput. Math. Math. Phys. 53 (6), 702–720 (2013)].

    MathSciNet  MATH  Google Scholar 

  17. S. A. Nazarov, “Enforced Stability of a Simple Eigenvalue in the Continuous Spectrum of aWaveguide,” Funktsional Anal. i Prilozhen. 47 (3), 37–53 (2013) [Funct. Anal. Appl. 47 (3), 195–209 (2013)].

    Article  MathSciNet  Google Scholar 

  18. R. R. Gadyl’shin, “Local Perturbations of the Schrödinger Operator on the Axis,” Teoret. Mat. Fiz. 132 (1), 97–104 (2002) [Theoret. and Math. Phys. 132 (1), 976–982 (2002)].

    Article  MathSciNet  MATH  Google Scholar 

  19. D. I. Borisov and R. R. Gadyl’shin, “On the Spectrum of a Periodic Operator with a Small Localized Perturbation,” Izv. Ross. Akad. Nauk Ser. Mat. 72 (4), 37–66 (2008) [Izv. Math. 72 (4), 659–688 (2008)].

    Article  MathSciNet  MATH  Google Scholar 

  20. E. Sanchez-Palencia, Non-Homogeneous Media and Vibration Theory (Springer, Berlin-Heidelberg, 1980; Mir, Moscow, 1984).

    MATH  Google Scholar 

  21. I. M. Glazman, Direct Methods of Qualitative Spectral Analysis of Singular Differential Operators (Fizmatlit, Moscow, 1963; Israel Program for Scientific Translations, Jerusalem, 1965; Daniel Davey & Co., Inc., New York, 1966).

    MATH  Google Scholar 

  22. T. Kato, Perturbation Theory for Linear Operators (Springer-Verlag New York, Inc., New York, 1966; Mir, Moscow, 1972).

    MATH  Google Scholar 

  23. V. P. Mikhailov, Partial Differential Equations (Nauka, Moscow, 1976).

    MATH  Google Scholar 

  24. B. V. Shabat, Introduction to Complex Analysis, Parts 1, 2 (Nauka, Moscow, 1976; Introduction a l’analyse complexe. Tome 1. Mir, Moscow, 1990; Part 2. Amer. Math. Soc., Providence, RI, 1992).

    MATH  Google Scholar 

  25. D. Borisov, P. Exner, and R. Gadyl’shin, “Geometric Coupling Thresholds in a Two-Dimensional Strip,” J. Math. Phys. 43 (12), 6265–6278 (2002).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  26. D. I. Borisov, “Discrete Spectrum of an Asymmetric Pair of Waveguides Coupled Through a Window,” Mat. Sb. 197 (4), 3–32 (2006) [Sb. Math. 197 (4), 475–504 (2006)].

    Article  MATH  Google Scholar 

  27. D. Krejcirik and P. Siegl, “PT-Symmetric Models in Curved Manifolds,” J. Phys. A 43 (48), id 485204 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  28. S. A. Nazarov, “Variational and Asymptotic Methods for Finding Eigenvalues below the Continuous Spectrum Threshold,” Sibirsk Mat. Zh. 51 (5), 1086–1101 (2010) [Siberian. Math. J. 51 (5), 866–878 (2010)].

    MathSciNet  Google Scholar 

  29. D. I. Borisov, “On a PT-Symmetric Waveguide with a Pair of Small Holes,” Tr. Inst. Mat. Mekh. (Ekaterinburg) 18 (2), 22–37 (2012). [Proc. Steklov Inst. Math. 281, Suppl. 1, S5–S21 (2013)].

    Google Scholar 

  30. S. A. Nazarov, “Asymptotics of an Eigenvalue on the Continuous Spectrum of Two Quantum Waveguides Coupled Through Narrow Windows,” Mat. Zametki 93 (2), 227–245 (2013) [Math. Notes 93 (2), 266–281 (2013)].

    Article  MathSciNet  MATH  Google Scholar 

  31. A. M. Il’in, Matching of Asymptotic Expansions of Solutions of Boundary Value Problems (Nauka, Moscow, 1989; Amer. Math. Soc., Providence, RI, 1992).

    MATH  Google Scholar 

  32. W. G. Mazja, S.A. Nasarow, and B. A. Plamenewski, Asymptotische Theorie Elliptischer Randwertaufgaben in Singulär Gestörten Gebieten, B. 1,2 (Akad.-Verl., Berlin, 1991).

    Google Scholar 

  33. I. Yu. Popov, “Asymptotics of Bound States for Laterally Coupled Waveguides,” Rep. Math. Phys. 43 (3), 427–437 (1999).

    Article  ADS  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Bashkir State Pedagogical University, ul. Oktyabr’skoi Revolyutsii, 3a, 450000, Ufa, Russia

    A. R. Bikmetov & R. R. Gadyl’shin

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  1. A. R. Bikmetov
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  2. R. R. Gadyl’shin
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Correspondence to A. R. Bikmetov.

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Bikmetov, A.R., Gadyl’shin, R.R. On local perturbations of waveguides. Russ. J. Math. Phys. 23, 1–18 (2016). https://doi.org/10.1134/S1061920816010015

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  • DOI: https://doi.org/10.1134/S1061920816010015

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