I would appreciate any answers or even references for the following problem.
Let $(M,g)$ be a complete smooth Riemannian manifold with an asymptotically Euclidean metric (let's even say that the metric is Euclidean outside some compact set). Let $\rho$ be a smooth positive function and consider the problem $$ -\Delta_g \phi = \lambda\, \rho\,\phi\quad \text{on $M$}.$$
I think it must be known that under suitable decay properties on $\rho$, there must be a complete basis of suitable weighted Sobolev spaces in terms a discrete set of eigenvalues and eigenfunctions $\{\phi_k,\lambda_k\}_{k\in \mathbb N}$ as above. Is this known in this general Riemannian setup?
Also if the answer is known, is an analogous Weyl's law known in the literature as well as quantitative eigenfunction bounds $\|\phi_k\|_{L^{p}_\delta(M)}$ as $k\to \infty$?
