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We prove the time decay estimates $L^1({\cal R}) \rightarrow L^\infty ({\calR}),$ where ${\cal R}$ is an infinite star-shaped network, for theSchr\"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 thesolution for initial conditions with a lower cutoff frequency tends to the freesolution, if the cutoff frequency tends to infinity.

Dispersive effects and high frequency behaviour for the Schr\"odinger equation in star-shaped networks

Dispersive effects and high frequency behaviour for the Schr\"odinger equation in star-shaped networks