Spectral analysis and stabilization of the dissipative Schr\"{o}dinger operator on the tadpole graph

TL;DR

This paper analyzes the spectral properties of the damped Schrödinger operator on a tadpole graph and proves exponential energy decay by establishing a Riesz basis of generalized eigenfunctions.

Contribution

It provides a detailed spectral analysis and demonstrates exponential decay for the damped Schrödinger semigroup on the tadpole graph, introducing a new basis construction.

Findings

Spectral analysis of the dissipative Schrödinger operator on the tadpole graph

Decomposition of the resolvent kernel

Exponential energy decay established

Abstract

We consider the damped Schr\"odinger semigroup $e^{-it \frac{d^2}{dx^2}}$ on the tadpole graph ${\mathcal R}$. We first give a careful spectral analysis and an appropriate decomposition of the kernel of the resolvent. As a consequence and by showing that the generalized eigenfunctions form a Riesz basis of some subspace of $L^2({\mathcal R})$, we prove that the corresponding energy decay exponentially.