Product formula related to quantum Zeno dynamics

TL;DR

This paper establishes a product formula for quantum Zeno dynamics involving a unitary group and a projection, providing insights into the limit behavior of quantum systems under frequent measurements.

Contribution

It proves a new product formula related to quantum Zeno dynamics with convergence results, especially in finite-dimensional subspaces, advancing understanding of quantum measurement effects.

Findings

Convergence in $L^2_{loc}$ for the product formula involving a semibounded operator and a projection.

Demonstration of convergence in the Hilbert space for finite-dimensional projections.

Application to free Schrödinger evolution with domain projections.

Abstract

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 $\HH$, with the convergence in $L^2_\mathrm{loc}(\mathbb{R};\HH)$. It gives a partial answer to the question about existence of the limit which describes quantum Zeno dynamics in the subspace \hbox{$\mathrm{Ran} P$}. The convergence in $\HH$ 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 Schr\"odinger evolution.