The Influence of the Tunnel Effect on L-time Decay

  • F. Ali MehmetiEmail author
  • R. Haller-Dintelmann
  • V. Régnier
Conference paper
Part of the Operator Theory: Advances and Applications book series (OT, volume 221)


We consider the Klein-Gordon equation on a star-shaped network composed of n half-axes connected at their origins.W e add a potential that is constant but different on each branch.E xploiting a spectral theoretic solution formula from a previous paper, we study the L -time decay via Hörmander’s version of the stationary phase method.W e analyze the coefficient c of the leading term \(c\cdot t^{ - {1 \mathord{\left/{\vphantom {1 2}} \right.\kern-\nulldelimiterspace}2}}\) of the asymptotic expansion of the solution with respect to time.F or two branches we prove that for an initial condition in an energy band above the threshold of tunnel effect, this coefficient tends to zero on the branch with the higher potential, as the potential difference tends to infinity. At the same time the incline to the t-axis and the aperture of the cone of \( t^{ - {1 \mathord{\left/{\vphantom {1 2}} \right.\kern-\nulldelimiterspace} 2}}\) -decay in the (t, x)-plane tend to zero.


Networks Klein-Gordon equation stationary phase method L-time decay 


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  1. 1.
    F. Ali Mehmeti, Spectral Theory and L -time Decay Estimates for Klein-Gordon Equations on Two Half Axes with Transmission: the Tunnel Effect. Math. Methods Appl. Sci. 17 (1994), 697–752.MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    F. Ali Mehmeti, Transient Waves in Semi-Infinite Structures: the Tunnel Effect and the Sommerfeld Problem. Mathematical Research, vol. 91, Akademie Verlag, Berlin, 1996.Google Scholar
  3. 3.
    F. Ali Mehmeti, R. Haller-Dintelmann, V. Régnier, Dispersive Waves with multiple tunnel effect on a star-shaped network; to appear in: “Proceedings of the Conference on Evolution Equations and Mathematical Models in the Applied Sciences (EEMMAS)”, Taranto 2009.Google Scholar
  4. 4.
    F. Ali Mehmeti, R. Haller-Dintelmann, V. Régnier, Multiple tunnel effect for dispersive waves on a star-shaped network: an explicit formula for the spectral representation. arXiv:1012.3068v1 [math.AP], preprint 2010.Google Scholar
  5. 5.
    F. Ali Mehmeti, V. Régnier, Delayed reflection of the energy flow at a potential step for dispersive wave packets. Math. Methods Appl. Sci. 27 (2004), 1145–1195.MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    J. von Below, J.A. Lubary, The eigenvalues of the Laplacian on locally finite networks. Results Math. 47 (2005), no. 3-4, 199–225.MathSciNetzbMATHGoogle Scholar
  7. 7.
    S. Cardanobile and D. Mugnolo. Parabolic systems with coupled boundary conditions. J. Differential Equations 247 (2009), no. 4, 1229–1248.MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Y. Daikh, Temps de passage de paquets d’ondes de basses fréquences ou limités en bandes de fréquences par une barrière de potentiel. Thèse de doctorat, Valenciennes, France, 2004.Google Scholar
  9. 9.
    J.M. Deutch, F.E. Low, Barrier Penetration and Superluminal Velocity. Annals of Physics 228 (1993), 184–202.MathSciNetCrossRefGoogle Scholar
  10. 10.
    A. Enders, G. Nimtz, On superluminal barrier traversal. J. Phys. I France 2 (1992), 1693–1698.CrossRefGoogle Scholar
  11. 11.
    A. Haibel, G. Nimtz, Universal relationship of time and frequency in photonic tunnelling. Ann. Physik (Leipzig) 10 (2001), 707–712.CrossRefGoogle Scholar
  12. 12.
    L. Hörmander, The Analysis of Linear Partial Differential Operators I. Springer, 1984.Google Scholar
  13. 13.
    V. Kostrykin, R. Schrader, The inverse scattering problem for metric graphs and the travelling salesman problem. Preprint, 2006 (
  14. 14.
    O. Liess, Decay estimates for the solutions of the system of crystal optics. Asymptotic Analysis 4 (1991), 61–95.MathSciNetzbMATHGoogle Scholar
  15. 15.
    B. Marshall, W. Strauss, S. Wainger, Lp -L q Estimates for the Klein-Gordon Equation. J. Math. Pures et Appl. 59 (1980), 417–440.MathSciNetzbMATHGoogle Scholar
  16. 16.
    K. Mihalincic, Time decay estimates for the wave equation with transmission and boundary conditions. Dissertation. Technische Universität Darmstadt, Germany, 1998.Google Scholar
  17. 17.
    M. Pozar, Microwave Engineering. Addison-Wesley, New York, 1990.Google Scholar

Copyright information

© Springer Basel 2012

Authors and Affiliations

  • F. Ali Mehmeti
    • 1
    • 2
    Email author
  • R. Haller-Dintelmann
    • 3
  • V. Régnier
    • 1
    • 2
  1. 1.Univ Lille Nord de FranceLilleFrance
  2. 2.UVHC, LAMAV, FR CNRS 2956ValenciennesFrance
  3. 3.Fachbereich MathematikTU DarmstadtDarmstadtGermany

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