Spectral estimates on the sphere

Abstract

In this article we establish optimal estimates for the first eigenvalue of Schrödinger operators on the $d$-dimensional unit sphere. These estimates depend on $L^p$ norms of the potential, or of its inverse, and are equivalent to interpolation inequalities on the sphere. We also characterize a semiclassical asymptotic regime and discuss how our estimates on the sphere differ from those on the Euclidean space.