Let $(M,g)$ be a closed Riemannian manifold and $\Delta$ be its Laplace-Beltrami operator. It is well known that the spectrum of $\Delta$ is a discrete subset of $[0,+\infty)$ and the eigenfunctions form an orthogonal basis of the Hilbert space $W^{1,2}(M)$. My question is:
Question: Given $a\in C^\infty(M)$, does the operator $Lf:=\Delta f+af$ have the same spectral property as $\Delta$ (except that the interval $[0,+\infty)$ should be replaced by $[\lambda_0,+\infty)$, where $\lambda_0$ is the lowest eigenvalue)?
My background on analysis is quite limited, so any hint about the ideas behind this kind of results and their generalizations would be appreciated!