  In this paper we consider the 1D Schr\"odinger operator $H$ with periodic
point interactions. We show an $L^1-L^\infty$ bound for the time evolution
operator $e^{-itH}$ restricted to each energy band with decay order
$O(t^{-1/3})$ as $t\to \infty$, which comes from some kind of resonant state.
The order $O(t^{-1/3})$ is optimal for our model. We also give an asymptotic
bound for the coefficient in the high energy limit. For the proof, we give an
asymptotic analysis for the band functions and the Bloch waves in the high
energy limit. Especially we give the asymptotics for the inflection points in
the graphs of band functions, which is crucial for the asymptotics of the
coefficient in our estimate.
