Existence of the spectral gap for elliptic operators

Let $M$ be a connected, noncompact, complete Riemannian manifold, consider the operator $L=\text{DD}+\text{nn}V$ for some $V\in C^2(M)$ with $\exp [V]$ integrable w.r.t. the Riemannian volume element. This paper studies the existence of the spectral gap of $L$. As a consequence of the main result, let $\text{rr}$ be the distance function from a point $o$, then the spectral gap exists provided ${\lim }_{\text{rr}\to \mathrm{\infty }}\sup L\text{rr}<0$ while the spectral gap does not exist if $o$ is a pole and ${\lim }_{\text{rr}\to \mathrm{\infty }}\inf L\text{rr}\ge 0.$ Moreover, the elliptic operators on ${\mathbb{R}}^d$ are also studied.