---
author:
- 'Pavel Exner$^{a,b}$ and Takashi Ichinose$^{c}$'
title: |
    A Product Formula Related to\
    Quantum Zeno Dynamics
---

> [*a) Department of Theoretical Physics, Nuclear Physics Institute,\
> Academy of Sciences, 25068 Řež, Czech Republic\
> b) Doppler Institute, Czech Technical University, Břehov[á]{} 7,\
> 11519 Prague, Czech Republic\
> c) Department of Mathematics, Faculty of Science, Kanazawa\
> University, Kanazawa 920-1192, Japan\
> exner@ujf.cas.cz, ichinose@kenroku.kanazawa-u.ac.jp*]{}
>
> [We prove a product formula which involves the unitary group generated by a semibounded self-adjoint operator and an orthogonal projection $P$ on a separable Hilbert space ${{\mathcal H}}$, with the convergence in $L^2_\mathrm{loc}({\mathbb{R}};{{\mathcal H}})$. It gives a partial answer to the question about existence of the limit which describes quantum Zeno dynamics in the subspace . The convergence in ${{\mathcal H}}$ is demonstrated in the case of a finite-dimensional $P$. The main result is illustrated in the example where the projection corresponds to a domain in ${\mathbb{R}}^d$ and the unitary group is the free Schrödinger evolution.]{}

Introduction
============

The fact that the decay of an unstable system can be slowed down, or even fully stopped in the ideal case, by frequently repeated measurements checking whether the system is still undecayed was 