Pavel Exner and Takashi Ichinose
Product formula related to quantum Zeno dynamics
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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. [a revised version of mp_arc 03-74,
to appear in Ann. H. Poincare]