  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.
  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.
