Advertisement

Journal of Evolution Equations

, Volume 12, Issue 3, pp 513–545 | Cite as

Multiple tunnel effect for dispersive waves on a star-shaped network: an explicit formula for the spectral representation

  • F. Ali MehmetiEmail author
  • R. Haller-Dintelmann
  • V. Régnier
Article

Abstract

We consider the Klein–Gordon equation on a star-shaped network composed of n half-axes connected at their origins. We add a potential that is constant but different on each branch. The corresponding spatial operator is self-adjoint, and we state explicit expressions for its resolvent and its resolution of the identity in terms of generalized eigenfunctions. This leads to a generalized Fourier-type inversion formula in terms of an expansion in generalized eigenfunctions. Further, we prove the surjectivity of the associated transformation, thus showing that it is in fact a spectral representation. The characteristics of the problem are marked by the non-manifold character of the star-shaped domain. Therefore, the approach via the Sturm–Liouville theory for systems is not well-suited. The considerable effort to construct explicit formulas involving the generalized eigenfunctions that incorporate the tunnel effect is justified for example by the perspective to study the influence of this effect on the L -time decay of solutions.

Mathematics Subject Classification

Primary 34B45 Secondary 42A38 47A10 47A60 47A70 

Keywords

Networks Spectral theory Resolvent Generalized eigenfunctions Functional calculus Evolution equations Dynamicsof the tunnel effect Klein–Gordon equation 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    F. Ali Mehmeti, Nonlinear Waves in Networks. Mathematical Research, vol. 80, Akademie Verlag, Berlin, (1994).Google Scholar
  2. 2.
    Ali Mehmeti F.: 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, 697–752 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    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
  4. 4.
    F. Ali Mehmeti, J. von Below, S. Nicaise (eds.), Partial differential equations on multistructures. Lecture Notes in Pure and Appl. Math., vol. 219, Marcel Dekker, New York, (2001).Google Scholar
  5. 5.
    Ali Mehmeti F., Haller-Dintelmann R., Régnier V.: Expansions in generalized eigenfunctions of the weighted Laplacian on star-shaped networks. In: Amann, H., Arendt, W., Hieber, M., Neubrander, F., Nicaise, S., Below, J. (eds) Functional analysis and Evolution Equations; The Günter Lumer Volume, pp. 1–16. Birkhäuser, Basel (2008)CrossRefGoogle Scholar
  6. 6.
    F. Ali Mehmeti, R. Haller-Dintelmann, V. Régnier, The influence of the tunnel effect on L time decay. W. Arendt, J. A. Ball, J. Behrndt, K.-H. Förster, V. Mehrmann, C. Trunk (eds): Spectral Theory, Mathematical System Theory, Evolution Equations, Differential and Difference Equations: IWOTA10; Springer, Basel; Operator Theory: Advances and Applications, 221 (2012), 11–24.Google Scholar
  7. 7.
    F. Ali Mehmeti, R. Haller-Dintelmann, V. Régnier, Energy flow above the threshold of tunnel effect. To appear in: Advances in Harmonic Analysis and Operator Theory, STOP 2011 Proceedings (the “Stefan Samko Anniversary Volume”). arXiv:1111.1140v1 [math.AP], Preprint LAMAV 11.12, Valenciennes, (2011).Google Scholar
  8. 8.
    Ali Mehmeti F., Meister E., Mihalinčić K.: Spectral Theory for the Wave Equation in Two Adjacent Wedges. Math. Methods Appl. Sci. 20, 1015–1044 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Ali Mehmeti F., Régnier V.: Splitting of energy of dispersive waves in a star-shaped network. Z. Angew. Math. Mech. 83(no. 2), 105–118 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Ali Mehmeti F., Régnier V.: Delayed reflection of the energy flow at a potential step for dispersive wave packets. Math. Methods Appl. Sci. 27, 1145–1195 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Ali Mehmeti F., Régnier V.: Global existence and causality for a transmission problem with a repulsive nonlinearity. Nonlinear Anal. 69, 408–424 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    von Below J., Lubary J.A.: The eigenvalues of the Laplacian on locally finite networks. Results Math. 47(no. 3-4), 199–225 (2005)MathSciNetzbMATHGoogle Scholar
  13. 13.
    J. M. Berezanskii, Expansions in eigenfunctions of selfadjoint operators. Transl. Math. Monogr., vol. 17, American Mathematical Society, Providence, (1968).Google Scholar
  14. 14.
    Cardanobile S., Mugnolo D.: Parabolic systems with coupled boundary conditions. J. Differential Equations 247(no. 4), 1229–1248 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Croc E., Dermenjian Y.: (Analyse spectrale d’une bande acoustique multistratifiée I : principe d’absorption limite pour une stratification simple). SIAM J. Math. Anal. 26(no. 24), 880–924 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    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
  17. 17.
    Deutch J.M., Low F.E.: Barrier Penetration and Superluminal Velocity. Annals of Physics 228, 184–202 (1993)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Dorn B.: Semigroups for flows in infinite networks. Semigroup Forum 76(no. 2), 341–356 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Dunford N., Schwartz J.T.: Linear Operators II. Wiley Interscience, New York (1963)zbMATHGoogle Scholar
  20. 20.
    Enders A., Nimtz G.: On superluminal barrier traversal. J. Phys. I France 2, 1693–1698 (1992)CrossRefGoogle Scholar
  21. 21.
    P. Exner, J. P. Keating, P. Kuchment, T. Sunada, A. Teplyaev (eds.), Analysis on Graphs and Its Applications. Proc. Sympos. Pure Math., vol. 77, AMS, 2008.Google Scholar
  22. 22.
    Haibel A., Nimtz G.: Universal relationship of time and frequency in photonic tunnelling. Ann. Physik (Leipzig) 10, 707–712 (2001)CrossRefGoogle Scholar
  23. 23.
    Heinzelmann G., Werner P.: Resonance phenomena in compound cylindrical waveguides. Math. Methods Appl. Sci. 29(no. 8), 877–945 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    V. Kostrykin, R. Schrader, The inverse scattering problem for metric graphs and the travelling salesman problem. Preprint, 2006 (http://www.arXiv.org:math.AP/0603010).
  25. 25.
    Kramar M., Sikolya E.: Spectral properties and asymptotic periodicity of flows in networks. Math. Z. 249, 139–162 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Poerschke T., Stolz G., Weidmann J.: Expansions in Generalized Eigenfunctions of Selfadjoint Operators. Math. Z. 202, 397–408 (1989)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Pozar M.: Microwave Engineering. Addison-Wesley, New York (1990)Google Scholar
  28. 28.
    Rudin W.: Real and Complex Analysis. McGraw-Hill Book Co., New York-Toronto, Ont.-London (1966)zbMATHGoogle Scholar
  29. 29.
    J. Weidmann, Spectral Theory of Ordinary Differential Operators. Lecture Notes in Mathematics, vol. 1258, Springer-Verlag, Berlin, (1987).Google Scholar

Copyright information

© Springer Basel AG 2012

Authors and Affiliations

  • F. Ali Mehmeti
    • 1
    Email author
  • R. Haller-Dintelmann
    • 2
  • V. Régnier
    • 1
  1. 1.UVHC, LAMAV, FR CNRS 2956ValenciennesFrance
  2. 2.TU Darmstadt, Fachbereich MathematikDarmstadtGermany

Personalised recommendations