Eigenvalue estimates for the weighted laplacian on a Riemannian manifold

Given a complete Riemannian manifold M and a smooth positive function w on M, let L = - Δ - ∇(log w) acting on L2(M, w dV). Generalizing techniques used in the case of the Laplacian, we obtain upper and lower bounds for the first non-zero eigenvalue of L, for M compact, and for the bottom of the spectrum, for M non-compact.