On the essential spectrum of the Laplacian and the drifted Laplacian

TL;DR

This paper investigates the essential spectrum of the Laplacian and drift Laplacian on weighted Riemannian manifolds, establishing conditions under which the spectrum is the entire non-negative real line.

Contribution

It proves that the essential spectrum is [0, +∞) under specific curvature and growth conditions, extending understanding of spectral properties in weighted manifolds.

Findings

Essential spectrum of drift Laplacian is [0, +∞) with nonnegative Bakry-Émery curvature and sublinear growth of f.

Essential spectrum of Laplacian is [0, +∞) when Ric_f ≥ 1/2 g and |∇f|^2 ≤ f.

f-volume growth estimates are crucial in the proofs and may be of independent interest.

Abstract

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.