ON SUMS OF EIGENVALUES OF ELLIPTIC OPERATORS ON MANIFOLDS

Ahmad El Soufi (1) , Evans Harrell (2) , Said Ilias (1) , Joachim Stubbe (3)

1 LMPT - Laboratoire de Mathématiques et Physique Théorique
2 School of Mathematics - Georgia Institute of Technology
3 Département de Mathématiques - EPFL

We use the averaged variational principle introduced in a recent article on graph spectra [7] to obtain upper bounds for sums of eigenvalues of several partial differential operators of interest in geometric analysis, which are analogues of Kröger 's bound for Neumann spectra of Laplacians on Euclidean domains [12]. Among the operators we consider are the Laplace-Beltrami operator on compact subdomains of manifolds. These estimates become more explicit and asymptotically sharp when the manifold is conformal to homogeneous spaces (here extending a result of Strichartz [21] with a simplified proof). In addition we obtain results for the Witten Laplacian on the same sorts of domains and for Schrödinger operators with confining potentials on infinite Euclidean domains. Our bounds have the sharp asymptotic form expected from the Weyl law or classical phase-space analysis. Similarly sharp bounds for the trace of the heat kernel follow as corollaries.

Mots clés: Manifold with density; Weighted Laplacian; Schr\"odinger operator; Witten Laplacian; Eigenvalue; Upper bound; phase space; Weyl law; homogeneous space; conformal

Domaines: Géométrie métrique [math.MG]; Théorie spectrale [math.SP]

Ahmad El Soufi, Evans Harrell, Said Ilias, Joachim Stubbe. ON SUMS OF EIGENVALUES OF ELLIPTIC OPERATORS ON MANIFOLDS. 2015. ⟨hal-01174814v1⟩