Feng-Yu Wang
\def\DD{\Delta} \def\dd{\delta} \def\vv{\varepsilon} \def\<{\langle} \def\>{\rangle} \def\d{\text{d}} \def\Ric{\text{Ric}} \def\nn{\nabla} \def\pp{\partial} \def\ff{\frac} \def\ss{\sqrt} \def\si{\sigma} \def\gg{\gamma} \def\ll{\lambda} \def\aa{\alpha} \def\bb{\beta} \def\Hess{\text{Hess}} \def\OO{\Omega } \def\e{\text{e}} \def\rr{\rho} \def\bl{\bar\lambda} Let $M$ be a connected, noncompact, complete Riemannian manifold, consider the operator $L=\DD +\nn V$ 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 $\rr$ be the distance function from a point $o$, then the spectral gap exists provided $\lim_{\rr\to\infty}\sup L\rr<0$ while the spectral gap does not exist if $o$ is a pole and $\lim_{\rr\to\infty}\inf L\rr\ge 0.$ Moreover, the elliptic operators on $\mathbb R^d$ are also studied.