  In this article we establish optimal estimates for the first eigenvalue of
Schr\"odinger operators on the d-dimensional unit sphere. These estimates
depend on Lebsgue's norms of the potential, or of its inverse, and are
equivalent to interpolation inequalities on the sphere. We also characterize a
semi-classical asymptotic regime and discuss how our estimates on the sphere
differ from those on the Euclidean space.

Analysis of PDEs