As a continuation of[13]where a Poincare type inequality was introduced to study the essential spectrum on the L2-space of a probability measure, this paper provides a modification of this inequality so that the infimum of the essential spectrum is well described even if the reference measure is infinite. High order eigenvalues as well as the corresponding semigroup are estimated by using this new inequality. Criteria of the inequality and estimates of the inequality constants are presented. Finally, some concrete examples are considered to illustrate the main results. In particular, estimates of high order eigenvalues obtained in this paper are sharp as checked by two examples on the Euclidean space.

In order to describe L2-convergence rates slower than exponential, the weak Poincar

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.