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We establish inequalities for the eigenvalues of Schr\"odinger operators oncompact submanifolds (possibly with nonempty boundary) of Euclidean spaces, ofspheres, and of real, complex and quaternionic projective spaces, which arerelated to inequalities for the Laplacian on Euclidean domains due to Payne,P\'olya, and Weinberger and to Yang, but which depend in an explicit way on themean curvature. In later sections, we prove similar results for Schr\"odingeroperators on homogeneous Riemannian spaces and, more generally, on anyRiemannian manifold that admits an eigenmap into a sphere, as well as for theKohn Laplacian on subdomains of the Heisenberg group. Among the consequences ofthis analysis are an extension of Reilly's inequality, bounding any eigenvalueof the Laplacian in terms of the mean curvature, and spectral criteria for theimmersibility of manifolds in homogeneous spaces.

Universal inequalities for the eigenvalues of Schrodinger operators on submanifolds