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

We consider the Klein?CGordon equation on a star-shaped network composed of n half-axes connected at their origins. We add a potential that 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?CLiouville theory for systems is not well-suited. The considerable effort to construct explicit formulas involving the generalized eigenfunctions that incorporate the tunnel effect is justified for example by the perspective to study the influence of this effect on the L ^??-time decay of solutions.