Given a nonnegative self-adjoint operator H acting on a separable Hilbert space and an orthogonal projection P such that \(H_P := (H^{1/2}P)^*(H^{1/2}P)\) is densely defined, we prove that \(\lim _{n\rightarrow \infty } (P\,\mathrm {e}^{-itH/n}P)^n = \mathrm {e}^{-itH_P}P\) 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 \(P(0)=P\).

中文翻译：

---

关于与Quantum Zeno Dynamics有关的产品公式的注释

给定一个非负自伴算子H作用于可分离的希尔伯特空间和正交投影P，使得\（H_P：=（H ^ {1/2} P）^ *（H ^ {1/2} P）\）严格定义，我们证明\（\ lim _ {n \ rightarrow \ infty}（P \，\ mathrm {e} ^ {-itH / n} P）^ n = \ mathrm {e} ^ {-itH_P} P \）在强算子拓扑中成立。当P被满足\（P（0）= P \）的强连续投影值函数替代时，我们还对该乘积公式进行了修改，并将其扩展为这种情况。