  Let $\Omega\subset\mathbb{R}^n$ be a domain, $\Gamma$ be a hyperplane
intersecting it. Let $\varepsilon>0$, and
$\Omega_\varepsilon=\Omega\setminus\overline{\Sigma_\varepsilon}$, where
$\Sigma_\varepsilon$ ("sieve") is an $\varepsilon$-neighbourhood of $\Gamma$
punctured by many narrow passages. When $\varepsilon\to0$, the number of
passages tends to infinity, while the diameters of their cross-sections tend to
zero. For the case of identical straight periodically distributed and
appropriately scaled passages T. Del Vecchio (1987) proved that the Neumann
Laplacian on $\Omega_\varepsilon$ converges in a strong resolvent sense to the
Laplacian on $\Omega\setminus\Gamma$ subject to the so-called
$\delta'$-conditions on $\Gamma$. We will refine this result by deriving
estimates on the rate of convergence in terms of various operator norms, and
providing the estimate for the distance between the spectra. The assumptions we
impose on the passages are rather general. For $n=2$ the results of T. Del
Vecchio are not complete, some cases remain as open problems, and in this work
we will fill these gaps.
