Universal inequalities for the eigenvalues of Schrodinger operators on submanifolds

We establish inequalities for the eigenvalues of Schr"odinger operators on compact submanifolds (possibly with nonempty boundary) of Euclidean spaces, of spheres, and of real, complex and quaternionic projective spaces, which are related 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 the mean curvature. In later sections, we prove similar results for Schr"odinger operators on homogeneous Riemannian spaces and, more generally, on any Riemannian manifold that admits an eigenmap into a sphere, as well as for the Kohn Laplacian on subdomains of the Heisenberg group. Among the consequences of this analysis are an extension of Reilly’s inequality, bounding any eigenvalue of the Laplacian in terms of the mean curvature, and spectral criteria for the immersibility of manifolds in homogeneous spaces.

💡 Research Summary

The paper investigates universal eigenvalue inequalities for Schrödinger operators of the form H = −Δ + q on compact submanifolds (possibly with boundary) embedded in Euclidean spaces, spheres, and real, complex, or quaternionic projective spaces. The authors extend classical results such as the Payne‑Pólya‑Weinberger (PPW) inequalities, Hile‑Protter (HP) bounds, and Yang’s inequalities, by incorporating an explicit curvature term involving the mean curvature vector h of the immersion.

The central result for submanifolds of ℝ^m (Theorem 2.1) provides three related inequalities. The first (I) is a quadratic bound
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