Dispersive effects and high frequency behaviour for the Schrödinger equation in star-shaped networks

Abstract

We prove the time decay estimates $L^1(\mathcal{R})\to L^{\infty }(\mathcal{R}),$ where $\mathcal{R}$ is an infinite star-shaped network, for the Schrödinger group $e^{it(-\frac{d^2}{d{x}^2}+V)}$ 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.