In terms of the equivalence of Poincare inequality and the existence of spectral gap, the super-Poincare inequality is suggested in the paper for the study of essential spectrum. It is proved for symmetric di

The weak Poincar

Under a curvature condition, we obtain some criteria of supercontractivity and ultracontractivity for (non-symmetric) di

Let (E;F;

Let M be a connected, noncompact, complete Riemannian manifold, consider the operator L=∆+∇V for some V ∈C2(M) with exp[V]integrable with respect to the Riemannian volume element. This paper studies the existence of the spectral gap of L. As a consequence of the main result, let be the distance function from a point o, then the spectral gap exists provided lim →∞ sup L<0 while the spectral gap does not existi fo is a pole and lim →∞ inf L≥0. Moreover, the elliptic operators on Rd are also studied.