Spectral estimates on the sphere

Jean Dolbeault (1) , Maria J. Esteban (1) , Ari Laptev (2)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
2 Department of Mathematics [Imperial College London]

In this article we establish optimal estimates for the first eigenvalue of Schrödinger 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.

Mots clés: Schrödinger operator; ground state; one bound state Keller-Lieb-Thirring inequality; Spectral problems; Partial differential operators on manifolds; Quantum theory; Estimation of eigenvalues; Sobolev inequality; interpolation; Gagliardo-Nirenberg-Sobolev inequalities; logarithmic Sobolev inequality

Domaines: Equations aux dérivées partielles [math.AP]

Jean Dolbeault, Maria J. Esteban, Ari Laptev. Spectral estimates on the sphere. 2013. ⟨hal-00770755v1⟩