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


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 


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


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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

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