  Let $\Omega\subset\mathbb{R}^n$ be a bounded domain. We perturb it to a
domain $\Omega^\varepsilon$ attaching a family of small protuberances with
"room-and-passage"-like geometry ($\varepsilon>0$ is a small parameter).
Peculiar spectral properties of Neumann problems in so perturbed domains were
observed for the first time by R. Courant and D. Hilbert. We study the case,
when the number of protuberances tends to infinity as $\varepsilon\to 0$ and
they are $\varepsilon$-periodically distributed along a part of
$\partial\Omega$. Our goal is to describe the behaviour of the spectrum of the
operator
$\mathcal{A}^\varepsilon=-(\rho^\varepsilon)^{-1}\Delta_{\Omega^\varepsilon}$,
where $\Delta_{\Omega^\varepsilon}$ is the Neumann Laplacian in
$\Omega^\varepsilon$, and the positive function $\rho^\varepsilon$ is equal to
$1$ in $\Omega$. We prove that the spectrum of $\mathcal{A}^\varepsilon$
converges as $\varepsilon\to 0$ to the "spectrum" of a certain boundary value
problem for the Neumann Laplacian in $\Omega$ with boundary conditions
containing the spectral parameter in a nonlinear manner. Its eigenvalues may
accumulate to a finite point.
