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

Abstract

We prove the time decay estimates L1(R)→L∞(R), where R is an infinite starshaped network, for the Schrödinger group eit(-d2/dx2+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.