  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. 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 tunnel effect generalized eigenfunctions is justified
for example by the perspective to study the influence of tunnel effect on the
L-infinity-time decay.
