Note on a Product Formula Related to Quantum Zeno Dynamics

TL;DR

This paper proves a new product formula related to quantum Zeno dynamics, showing the strong operator limit of repeated projections and evolutions converges to a modified exponential, with extensions to time-dependent projections.

Contribution

It establishes a novel product formula for quantum Zeno dynamics involving unbounded operators and extends it to time-dependent projections.

Findings

Proved the limit of the product involving projections and exponentials converges strongly.

Derived modifications of the product formula for different projection scenarios.

Extended the formula to time-dependent projection-valued functions.

Abstract

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 $P(0)=P$.