  We investigate tensor products of Hilbert complexes, in particular the
(essential) spectrum of their Laplacians. It is shown that the essential
spectrum of the Laplacian associated to the tensor product complex is
computable in terms of the spectra of the factors. Applications are given for
the $\overline\partial$-Neumann problem on the product of two or more Hermitian
manifolds, especially regarding (non-) compactness of the associated
$\overline\partial$-Neumann operator.
