The Klein-Gordon equation with multiple tunnel effect on a star-shaped network: Expansions in generalized eigenfunctions

TL;DR

This paper analyzes the Klein-Gordon equation on a star-shaped network with multiple tunneling effects, providing explicit spectral decompositions and generalized eigenfunction expansions for the associated operator.

Contribution

It introduces explicit resolvent and spectral resolution formulas for the Klein-Gordon operator on a star-shaped network with branch-dependent potentials, using generalized eigenfunctions.

Findings

Explicit resolvent expressions derived

Generalized Fourier expansion established

Spectral properties characterized for the network operator

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. 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.