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 $\int_M h dv_g > 0$. Does it follow that the differential operator $\Delta + c h$ is positive (i.e., has a positive principal eigenvalue), where $c$ is any positive real number?

Let $(M,g)$ be a closed, smooth Riemannian manifold. Let $\Delta = -div\nabla$ be the Laplace-Beltrami operator. Let $h$ be a smooth function on $M$. Is there a condition on $h$ weaker than non-negativity such that $\Delta + h$ is a positive or non-negative operator?

I'm thinking of something akin to the following: For the conformal Laplacian, non-negativity of the Yamabe constant is sufficient for any $h$ that is a scalar curvature of a metric in the conformal class.