  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.
