particle in a domain of $\Omega
\subset{\mathbb{R}}^d$ is repeatedly ascertained. Using the method of stationary phase the authors showed that the Zeno dynamics describes in this case the free particle confined to $\Omega$, with the hard-wall (Dirichlet) condition at the boundary of the domain. The result cannot be regarded as fully rigorous, because detailed properties of the convergence are not worked out, but the idea is sound without any doubt.

In the present paper we combine the results of [@Ch1; @Ch2] with that of Kato [@Ka2] to address this question in a general setting. We show that if the natural effective Hamiltonian mentioned above is densely defined — which is a nontrivial assumption — then the Zeno dynamics exists and the said operator is its generator in a topology which includes an averaging over the time variable – cf. Theorem \[main\] for exact statement (a part of the present result given in Corollary \[cor2\] was announced in [@EI]). Our conclusion cannot be thus regarded as fully satisfactory from the mathematical point of view, because the natural topology to be used here is given by the norm of the Hilbert space, and in this respect an important part of the problem 