  We analyze the spectral properties of a particular class of unbounded open
sets. These are made of a central bounded ``core'', with finitely many
unbounded tubes attached to it. We adopt an elementary and purely variational
point of view, studying the compactness (or the defect of compactness) of level
sets of the relevant constrained Dirichlet integral. As a byproduct of our
argument, we also get exponential decay at infinity of variational
eigenfunctions. Our analysis includes as a particular case a planar set
(sometimes called ``bookcover''), already encountered in the literature on
curved quantum waveguides. J. Hersch suggested that this set could provide the
sharp constant in the {\it Makai-Hayman inequality} for the bottom of the
spectrum of the Dirichlet-Laplacian of planar simply connected sets. We
disprove this fact, by means of a singular perturbation technique.
