  We consider here a family of singular Laplace-Beltrami operators, focussing
our attention on the problem of so-called quantum confinement on the half-plane
equipped with Riemannian metrics of Grushin type degenerate at the boundary. By
introducing a costant-fiber direct integral scheme we are able to rigorously
characterize the presence or absence of self-adjointness of these operators. We
also compare our technique and results with the already studied problem of
quantum confinement on the half-cylinder.
