Energy Flow Above the Threshold of Tunnel Effect

  • F. Ali Mehmeti
  • R. Haller-Dintelmann
  • V. Régnier
Conference paper
Part of the Operator Theory: Advances and Applications book series (OT, volume 229)

Abstract

We consider the Klein-Gordon equation on two half-axes connected at their origins. We add a potential that is constant but different on each branch. In a previous paper, we studied the L-time decay via Hörmander’s version of the stationary phase method. Here we apply these results to show that for initial conditions in an energy band above the threshold of the tunnel effect a fixed portion of the energy propagates between group lines. Further we consider the situation that the potential difference tends to infinity while the energy band of the initial condition is shifted upwards such that the particle stays above the threshold of the tunnel effect. We show that the total transmitted energy as well as the portion between the group lines tend to zero like \( {{a_{2}}^{-1/2}} \)in the branch with the higher potential a2 as a2 tends to infinity. At the same time the cone formed by the group lines inclines to the t-axis while its aperture tends to zero.

Keywords

Networks Klein-Gordon equation stationary phase method L -time decay energy flow. 

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Copyright information

© Springer Basel 2013

Authors and Affiliations

  • F. Ali Mehmeti
    • 1
    • 2
  • 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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