Periodic coherent states decomposition and quantum dynamics on the flat torus

We provide a result on the coherent states decomposition for functions in $L^2 (\Bbb T^n)$ where $\Bbb T^n := (\Bbb R / 2\pi \Bbb Z)^n$. Suddenly, we study such a decomposition with respect to the quantum dynamics related to semiclassical elliptic Pseudodifferential operators, and we prove a related invariance result.