Dispersive effects for the Schrödinger equation on the tadpole graph

We consider the free Schrödinger group e−itd2dx2 on the tadpole graph R. We first show that the time decay estimates L1(R)→L∞(R) is in |t|−12 with a constant independent of the circumference 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.