Universal bounds for eigenvalues of Schrodinger operator on Riemannian manifolds

lit this paper we consider eigenvalues of Schrodinger operator with a weight on compact Riemannian manifolds with boundary (possibly empty) and prove a general inequality for them. By using this inequality, we study eigenvalues of Schriodinger operator with a weight on compact domains in a unit sphere, a complex projective space and a minimal submanifold in a Euclidean space. We also study the same problem on closed minimal submanifolds in a sphere, compact homogeneous space and closed complex hypersurfaces in a complex projective space. We give explict bound for the (k + 1)-th eigenvalue of the Schrodinger operator on such objects in terms of its first k eigenvalues. Our results generalize many previous estimates on eigenvalues of the Laplacian.