This is mostly enhancing Nik Weaver's comments.  Suppose  that $M$ is compact of dimension $m$. If $m\geq 2$, then for a generic metric $g$ on $M$ the eigenvalues $\lambda_k$  of the Laplacian $\Delta_g$ are   simple. In general, for any $m$,  Weyl's spectral estimates  imply that

$$\lambda_k \sim C_m \left(\frac{k}{{\rm vol}_g(M)}\right)^{\frac{2}{m}}\;\;\mbox{as $k\to\infty$},  $$

where $C_m$ is an explicit universal constant that depends only on $m$. (*Hat-tip to Marc Palm!*)  In particular this shows that for $m\geq 2$ and a generic metric  we have

$$0<\lambda_{k+1}-\lambda_k =O(1). $$

Now Nik Weaver's argument shows that there exists $r_0>0$ such   for any $\xi\in [0,\infty)\setminus {\rm spec}\;(\Delta)$  we have  $\Vert(R(\xi)\Vert\geq r_0$.