On the essential spectrum of the Laplacian and the drifted Laplacian

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