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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)

Abstract

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.

Keywords

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

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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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