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 $\mathrm{\hslash }\to 0$ and large quantum number $N\to \mathrm{\infty }$, with $\mathrm{\hslash }N$ 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.