Multiple tunnel effect for dispersive waves on a star-shaped network: an explicit formula for the spectral representation
Abstract
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.
Cite
@article{arxiv.1012.3068,
title = {Multiple tunnel effect for dispersive waves on a star-shaped network: an explicit formula for the spectral representation},
author = {Felix Ali Mehmeti and Robert Haller-Dintelmann and Virginie Régnier},
journal= {arXiv preprint arXiv:1012.3068},
year = {2010}
}
Comments
This article is a substantially extended version of the paper "The Klein-Gordon equation with multiple tunnel effect on a star-shaped network: Expansions in generalized eigenfunctions" (arXiv:0906.3230v1 [math.SP])