  This paper concerns the $L^2$ essential spectrum of the Laplacian $\Delta$
and the drift Laplacian $\Delta_f$ on complete Riemannian manifolds endowed
with a weighted measure $e^{-f}d\;vol_g$. We prove that the essential spectrum
of the drift Laplacian $\Delta_f$ is $[0,+\infty) $ provided the Bakry-\'Emery
curvature tensor $Ric_f$ is nonnegative and $f$ has sublinear growth . When
$Ric_f \geq 1/2 g$ and $|\nabla f|^2 \leq f$, we show that the essential
spectrum of the Laplacian is also $[0,+\infty)$. During the proofs of these
results, the $f$-volume growth estimate plays an important role and may be of
independent interest.

Differential Geometry