Product formula related to quantum Zeno dynamics

Details Product formula related to quantum Zeno dynamics Product formula related to quantum Zeno dynamics

2003-02-25

arxiv.org/abs/math-ph/0302060v2

Ann. H. Poincare 6 (2005), 195-215

Physics

Mathematical Physics

LaTeX 2e, 24 pages, with substantial modifications, to appear in Ann. H. Poincare

Scientific paper

10.1007/s00023-005-0203-2

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.

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