On the spectrum of Robin Laplacian in a planar waveguide

  • Alex Ferreira RossiniEmail author


We consider the Laplace operator in a planar waveguide, i.e. an infinite two-dimensional straight strip of constant width, with Robin boundary conditions. We study the essential spectrum of the corresponding Laplacian when the boundary coupling function has a limit at infinity. Furthermore, we derive sufficient conditions for the existence of discrete spectrum.


planar waveguide discrete spectrum Robin boundary conditions 


47B25 47F05 49R05 81Q10 


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  1. [1]
    R. A. Adams: Sobolev Spaces. Pure and Applied Mathematics 65, Academic Press, New York, 1975.Google Scholar
  2. [2]
    D. Borisov, G. Cardone: Planar waveguide with “twisted” boundary conditions: Discrete spectrum. J. Math. Phys. 52 (2011), 123513, 24 pages.MathSciNetCrossRefGoogle Scholar
  3. [3]
    D. Borisov, D. Krejcirík: PT-symmetric waveguides. Integral Equations Oper. Theory 62 (2008), 489–515.CrossRefGoogle Scholar
  4. [4]
    B. Chenaud, P. Duclos, P. Freitas, D. Krejcirík: Geometrically induced discrete spectrum in curved tubes. Differ. Geom. Appl. 23 (2005), 95–105.MathSciNetCrossRefGoogle Scholar
  5. [5]
    C. R. de Oliveira: Intermediate Spectral Theory and Quantum Dynamics. Progress in Mathematical Physics 54, Birkhäuser, Basel, 2009.CrossRefGoogle Scholar
  6. [6]
    C. R. de Oliveira, A. A. Verri: On the spectrum and weakly effective operator for Dirichlet Laplacian in thin deformed tubes. J. Math. Anal. Appl. 381 (2011), 454–468.MathSciNetCrossRefGoogle Scholar
  7. [7]
    J. Dittrich, J. Kríž: Bound states in straight quantum waveguides with combined boundary conditions. J. Math. Phys. 43 (2002), 3892–3915.MathSciNetCrossRefGoogle Scholar
  8. [8]
    P. Duclos, P. Exner: Curvature-induced bound states in quantum waveguides in two and three dimensions. Rev. Math. Phys. 7 (1995), 73–102.MathSciNetCrossRefGoogle Scholar
  9. [9]
    L. C. Evans: Partial Differential Equations. Graduate Studies in Mathematics 19, AMS, Providence, 1998.Google Scholar
  10. [10]
    P. Exner, P. Seba: Bound states in curved quantum waveguides. J. Math. Phys. 30 (1989), 2574–2580.MathSciNetCrossRefGoogle Scholar
  11. [11]
    P. Freitas, D. Krejcirík: Waveguides with combined Dirichlet and Robin boundary conditions. Math. Phys. Anal. Geom. 9 (2006), 335–352.MathSciNetCrossRefGoogle Scholar
  12. [12]
    J. Goldstone, R. L. Jaffe: Bound states in twisting tubes. Phys. Rev. B. 45 (1992), 14100–14107. doiCrossRefGoogle Scholar
  13. [13]
    M. Jílek: Straight quantum waveguide with Robin boundary conditions. SIGMA, Symmetry Integrability Geom. Methods Appl. 3 (2007), Paper 108, 12 pages.zbMATHGoogle Scholar
  14. [14]
    D. Krejcirík: Spectrum of the Laplacian in a narrow curved strip with combined Dirichlet and Neumann boundary conditions. ESAIM, Control Optim. Calc. Var. 15 (2009), 555–568.MathSciNetCrossRefGoogle Scholar
  15. [15]
    D. Krejcirík, J. Kríz: On the spectrum of curved planar waveguides. Publ. Res. Inst. Math. Sci. 41 (2005), 757–791.MathSciNetCrossRefGoogle Scholar
  16. [16]
    O. Olendski, L. Mikhailovska: Theory of a curved planar waveguide with Robin boundary conditions. Phys. Rev. E. 81 (2010), 036606. doiCrossRefGoogle Scholar
  17. [17]
    M. Reed, B. Simon: Methods of Modern Mathematical Physics. IV: Analysis of Operators. Academic Press, New York, 1978.zbMATHGoogle Scholar

Copyright information

© Institute of Mathematics of the Academy of Sciences of the Czech Republic, Praha, Czech Republic 2018

Authors and Affiliations

  1. 1.Instituto de MatemáticaUniversidade Federal de Mato Grosso do Sul, Cidade UniversitáriaCampo GrandeBrazil

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