Dispersive effects for the Schr\"odinger equation on a tadpole graph

We consider the free Schr\"odinger group $e^{-it \frac{d^2}{dx^2}}$ on a tadpole graph ${\mathcal R}$. We first show that the time decay estimates $L^1 ({\mathcal R}) \rightarrow L^\infty ({\mathcal R})$ is in $|t|^{-\frac12}$ with a constant independent of the length of the circle. Our proof is based on an appropriate decomposition of the kernel of the resolvent. Further we derive a dispersive perturbation estimate, which proves that the solution on the queue of the tadpole converges uniformly, after compensation of the underlying time decay, to the solution of the Neumann half-line problem, as the circle shrinks to a point. To obtain this result, we suppose that the initial condition fulfills a high frequency cutoff.