An addition theorem for the manifolds with the Laplacian having discrete spectrum

The question of the preservation of discreteness of the spectrum of the Laplacian acting in a space of differential forms under the cutting and gluing of manifolds reduces to the same problem for compact solvability of the operator of exterior derivation. Along these lines, we give some conditions on a cut $Y$ dividing a Riemannian manifold $X$ into two parts $X_+$ and $X_-$ under which the spectrum of the Laplacian on $X$ is discrete if and only if so are the spectra of the Laplacians on $X_+$ and $X_-$.