The Influence of the Tunnel Effect on L∞-time Decay

We consider the Klein-Gordon equation on a star-shaped network composed of n half-axes connected at their origins. We add a potential which is constant but different on each branch. Exploiting a spectral theoretic solution formula from a previous paper, we study the L∞-time decay via Hörmander’s version of the stationary phase method. We analyze the coefficient c of the leading term c ·t−1/2 of the asymptotic expansion of the solution with respect to time. For 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−1/2-decay in the (t, x)-plane tend to zero.