The semiclassical limit of a quantum Zeno dynamics

Motivated by a quantum Zeno dynamics in a cavity quantum electrodynamics setting, we study the asymptotics of a family of symbols corresponding to a truncated momentum operator, in the semiclassical limit of vanishing Planck constant h -> 0 and large quantum number N -> infinity, with hN kept fixed. In a suitable topology, the limit is the discontinuous symbol p chi(D) (x, p) where chi(D) is the characteristic function of the classically permitted region D in phase space. A refined analysis shows that the symbol is asymptotically close to the function p chi((N))(D) (x, p), where chi((N))(D) is a smooth version of chi(D) related to the integrated Airy function. We also discuss the limit from a dynamical point of view.