Remarks on the Cwikel-Lieb-Rozenblum and Lieb-Thirring Estimates for Schrödinger Operators on Riemannian Manifolds

El Maati Ouhabaz (1)

1 IMB - Institut de Mathématiques de Bordeaux

Let M be a general complete Riemannian manifold and consider a Schrödinger operator − + V on L2(M). We prove Cwikel-Lieb-Rozenblum as well as Lieb-Thirring type estimates for − + V . These estimates are given in terms of the potential and the heat kernel of the Laplacian on the manifold. Some of our results hold also for Schrödinger operators with complex-valued potentials.

Mots clés: Spectral theory; Schrödinger operator; Cwikel-Lieb-Rozenblum estimates; Lieb-Thirring estimates; Riemannian manifolds

Domaines: Equations aux dérivées partielles [math.AP]; Analyse fonctionnelle [math.FA]

El Maati Ouhabaz. Remarks on the Cwikel-Lieb-Rozenblum and Lieb-Thirring Estimates for Schrödinger Operators on Riemannian Manifolds. Acta Applicandae Mathematicae, 2010, 110, pp.1449-1459. ⟨10.1007/s10440-009-9519-0⟩. ⟨hal-00998106⟩