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

TL;DR

This paper derives explicit spectral formulas for the Klein-Gordon equation on a star-shaped network with different potentials on each branch, highlighting the tunnel effect's role in wave decay analysis.

Contribution

It provides explicit resolvent and spectral representation formulas for the Klein-Gordon operator on a star-shaped network with variable potentials, addressing non-manifold domain challenges.

Findings

Explicit formulas for resolvent and spectral measures

Spectral representation established via generalized eigenfunctions

Foundation for studying tunnel effect impact on wave decay

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.