Universal inequalities for the eigenvalues of Schrodinger operators on submanifolds

TL;DR

This paper derives universal eigenvalue inequalities for Schrödinger operators on various submanifolds, linking spectral properties to geometric features like mean curvature across different ambient spaces.

Contribution

It introduces explicit eigenvalue inequalities for Schrödinger operators on submanifolds in diverse geometries, extending classical Laplacian bounds and connecting spectral data with geometric immersibility.

Findings

Extended Reilly's inequality involving mean curvature

Derived eigenvalue bounds depending on mean curvature

Established spectral criteria for manifold immersibility

Abstract

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.