Contents 
We consider KleinGordon Equations with piecewise constant coefficients on a starshaped network with semiinfinite branches.
Using the spectral theory of the associated spatial operator developed in and a version of the stationary phase method, we calculate the leading term of an asymptotic expansion for large times including an error estimate for solutions in frequency bands. This yields the exact time decay of the solution inside the cone of group lines issued by the frequency band on the outgoing half axis.
The special case of two branches can be interpreted in physical terms for example as a simplified model for two connected semi infinite electromagnetic waveguides with different dielectric constants. In this case we show that the ratio of the energy on the outgoing half axis and inside the cone of group lines is bounded by a constant which is essentially independent of the height of the potential step but is determined by the frequency band.
This suggests that the particle character of a wave packet is quite independent of the potentials with which it interacts, but depends mainly on its frequency band. 
