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

We consider the damped Schr"odinger semigroup $e^{-it\frac{d^2}{d{x}^2}}$ 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.