Let $(M,g)$ be a closed, smooth Riemannian manifold. Let $\Delta = -div\nabla$ be the Laplace-Beltrami operator. Let $h$ be a smooth function such that on $\int_M h dv_g > 0$. Does it follow M$. Is there a condition on $h$ weaker than non-negativity such that the differential operator $\Delta + c h$ is positive (i.e., has a positive principal eigenvalue)or non-negative operator?
I'm thinking of something akin to the following: For the conformal Laplacian, where $c$ non-negativity of the Yamabe constant is sufficient for any positive real number?$h$ that is a scalar curvature of a metric in the conformal class.