In this paper the linear stability properties of the steady states of a no-slip lubrication equation are studied. In the physical context, these steady states correspond to configurations of droplets that arise during the late-phase dewetting process under the influence of both destabilizing van der Waals and stabilizing Born intermolecular forces, which in turn give rise to the minimum thickness $\varepsilon$ of the remaining film connecting the droplets. The goal of this paper is to give an asymptotic description of the eigenvalues and eigenfunctions of the problem, linearized about the one-droplet solutions, as $\varepsilon\to0$. For this purpose, corresponding asymptotic eigenvalue problems with piecewise constant coefficients are constructed such that their eigenvalue asymptotics can be determined analytically. A comparison with numerically computed eigenvalues and eigenfunctions shows good agreement with the asymptotic results and the existence of a spectrum gap for sufficiently small $\varepsilon$.