Existence of the spectral gap for elliptic operators
Feng-Yu Wang
Let $M$ be a connected, noncompact, complete Riemannian manifold, consider the operator $L=\Delta +\nabla V$ 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 $\rho$ be the distance function from a point $o$, then the spectral gap exists provided $\lim_{\rho\to\infty}\sup L\rho \lt 0$ while the spectral gap does not exist if $o$ is a pole and $\lim_{\rho\to\infty}\inf L\rho\ge 0.$ Moreover, the elliptic operators on $\mathbb R^d$ are also studied.