Product formula related to quantum Zeno dynamics

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.Comment: LaTeX 2e, 24 pages, with substantial modifications, to appear in Ann. H. Poincar