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 $\hbar\to0$ and large quantum number $N\to\infty$, with $\hbar N$ 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 $p\chi_D^{(N)}(x,p)$, where $\chi_D^{(N)}$ is a smooth version of $\chi_D$ related to the integrated Airy function. We also discuss the limit from a dynamical point of view.