Universal inequalities for the eigenvalues of Schrodinger operators on submanifolds

Ahmad El Soufi (1) , Evans Harrell (2) , Said Ilias (1)

1 LMPT - Laboratoire de Mathématiques et Physique Théorique
2 School of Mathematics - Georgia Institute of Technology

We establish inequalities for the eigenvalues of Schr\"{o}-dinger 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, Pólya, 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\"{o}dinger 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.

Mots clés: eigenvalues; Laplacian; Schr\"{o}dinger operator; Reilly inequality; Kohn Laplacian

Domaines: Géométrie métrique [math.MG]

Ahmad El Soufi, Evans Harrell, Said Ilias. Universal inequalities for the eigenvalues of Schrodinger operators on submanifolds. Transactions of the American Mathematical Society, 2009, 361 (5), pp.2337--2350. ⟨hal-00360706⟩