  For a bounded corner domain $\Omega$, we consider the Robin Laplacian in
$\Omega$ with large Robin parameter. Exploiting multiscale analysis and a
recursive procedure, we have a precise description of the mechanism giving the
ground state of the spectrum. It allows also the study of the bottom of the
essential spectrum on the associated tangent structures given by cones. Then we
obtain the asymptotic behavior of the principal eigenvalue for this singular
limit in any dimension, with remainder estimates. The same method works for the
Schr\"odinger operator in $\mathbb{R}^n$ with a strong attractive
delta-interaction supported on $\partial\Omega$. Applications to some Erhling's
type estimates and the analysis of the critical temperature of some
superconductors are also provided.
