Note on a Product Formula Related to Quantum Zeno Dynamics

Given a nonnegative self-adjoint operator $H$ acting on a separable Hilbert space and an orthogonal projection $P$ such that $H_P := (H^{1/2}P)^*(H^{1/2}P)$isdenselydefined,weprovethat$\lim_{n\rightarrow \infty} (P,$\mathrm{e}$^{-itH/n}P)^n = $\mathrm{e}$^{-itH_P}P$ holds in the strong operator topology. We also derive modifications of this product formula and its extension to the situation when $P$ is replaced by a strongly continuous projection-valued function satisfying $P(0)=P$.