Spectral analysis and best decay rate of the wave propagator on the tadpole graph

TL;DR

This paper performs a detailed spectral analysis of the damped wave operator on a tadpole graph, establishing the optimal exponential decay rate of energy through spectral and eigenfunction analysis.

Contribution

It provides a comprehensive spectral decomposition and proves the Riesz basis property of eigenfunctions, leading to the determination of the optimal decay rate.

Findings

Spectral analysis of the damped wave operator on the tadpole graph

Decomposition of the resolvent's kernel

Establishment of exponential energy decay with optimal rate

Abstract

We consider the damped wave semigroup on the tadpole graph ${\mathcal R}$. We first give a meticulous spectral analysis, followed by a judicious decomposition of the resolvent's kernel. As a consequence, and by showing that the generalized eigenfunctions form a Riesz basis of some subspace of the energy space $\mathcal{H}$, we establish the exponential decay of the corresponding energy, with the optimal decay rate dictated by the spectral abscissa of the relevant operator.