Mathematics – Analysis of PDEs
Scientific paper
2010-12-14
Mathematics
Analysis of PDEs
This article is a substantially extended version of the paper "The Klein-Gordon equation with multiple tunnel effect on a star
Scientific paper
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.
Haller-Dintelmann Robert
Mehmeti Felix Ali
Régnier Virginie
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