Let $M$ be a closed (compact without boundary) Riemannian manifold. Is there a body of results that compares the eigenvalues of the Laplace-Beltrami operator with that of Schrödinger operators $-\Delta + V$? To make it more specific, a particular Schrödinger operator that is well-studied is $L = -\Delta + cR$, where $R$ is the scalar curvature of the manifold and $c$ is a positive constant, with $c = \frac{1}{2}$ or $c = \frac{1}{4}$ being two popular choices for $c$. Is there any way to compare the spectrum of $L$ and $\Delta$? At least in the special case of surfaces, or manifolds of nice curvature properties? I realize that the question is very broad-ranged and vague, but this is mainly a reference request. Thanks!
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