  We investigate the operator $-\Delta -\alpha \delta (x-\Gamma)$ in
$L^2(\mathbb{R}^3)$, where $\Gamma$ is a smooth surface which is either compact
or periodic and satisfies suitable regularity requirements. We find an
asymptotic expansion for the lower part of the spectrum as $\alpha\to\infty$
which involves a ``two-dimensional'' comparison operator determined by the
geometry of the surface $\Gamma$. In the compact case the asymptotics concerns
negative eigenvalues, in the periodic case Floquet eigenvalues. We also give a
bandwidth estimate in the case when a periodic $\Gamma$ decomposes into compact
connected components. Finally, we comment on analogous systems of lower
dimension and other aspects of the problem.
