JournalsjstVol. 7, No. 4pp. 985–1022

On sums of eigenvalues of elliptic operators on manifolds

  • Ahmad El Soufi

    Université de Tours, France (†)
  • Evans M. Harrell II

    Georgia Institute of Technology, Atlanta, USA
  • Saïd Ilias

    Université de Tours, France
  • Joachim Stubbe

    EPFL, Lausanne, Switzerland
On sums of eigenvalues of elliptic operators on manifolds cover
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Abstract

We use the averaged variational principle introduced in a recent article on graph spectra [10] 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 [15]. 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 [26] with a simplied 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.

Cite this article

Ahmad El Soufi, Evans M. Harrell II, Saïd Ilias, Joachim Stubbe, On sums of eigenvalues of elliptic operators on manifolds. J. Spectr. Theory 7 (2017), no. 4, pp. 985–1022

DOI 10.4171/JST/183