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We consider the Klein-Gordon equation on a star-shaped network composed of nhalf-axes connected at their origins. We add a potential which is constant butdifferent on each branch. The corresponding spatial operator is self-adjointand we state explicit expressions for its resolvent and its resolution of theidentity in terms of generalized eigenfunctions. This leads to a generalizedFourier type inversion formula in terms of an expansion in generalizedeigenfunctions. Further we prove the surjectivity of the associatedtransformation, thus showing that it is in fact a spectral representation. Thecharacteristics of the problem are marked by the non-manifold character of thestar-shaped domain. Therefore the approach via the Sturm-Liouville theory forsystems is not well-suited. The considerable effort to construct explicitformulas involving the tunnel effect generalized eigenfunctions is justifiedfor example by the perspective to study the influence of tunnel effect on theL-infinity-time decay.

Multiple tunnel effect for dispersive waves on a star-shaped network: an explicit formula for the spectral representation