The semiclassical limit of a quantum Zeno dynamics

Abstract

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 \(\hbar \rightarrow 0\) and large quantum number \(N\rightarrow \infty \), 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.