  We prove the time decay estimates $L^1({\cal R}) \rightarrow L^\infty ({\cal
R}),$ where ${\cal R}$ is an infinite star-shaped network, for the
Schr\"odinger group $e^{it(- \frac{d^2}{dx^2} + V)}$ for real-valued potentials
$V$ satisfying some regularity and decay assumptions. Further we show that the
solution for initial conditions with a lower cutoff frequency tends to the free
solution, if the cutoff frequency tends to infinity.
