A logarithmic improvement in the two-point Weyl Law for manifolds without conjugate points
[Une amélioration logarithmique de la loi de Weyl à deux-points sur les variétés sans points conjugués]
Keeler, Blake1
1 McGill University, 805 Sherbrooke St. W., Montreal QC (Canada)
Annales de l'Institut Fourier, Online first, 44 p.

Dans cet article, nous étudions la loi de Weyl à deux points pour l’opérateur de Laplace–Beltrami sur une variété riemannienne lisse et compacte M sans points conjugués. Cela veut dire que nous donnons le comportement asymptotique du noyau de Schwartz, E λ (x,y), de l’opérateur de projection de L 2 (M) sur la somme directe des espaces propres pour des valeurs propres inférieures à λ 2 quand λ→∞. Dans le régime où x,y sont restreints à un voisinage compact de la diagonale en M×M, on obtient une amélioration logarithmique uniforme dans le reste du développement asymptotique pour E λ et ses dérivées de tous ordres , ce qui généralise un résultat de Bérard, qui a traité le cas diagonal E λ (x,x). Lorsque x,y évitent un voisinage compact de la diagonale, on obtient cette même amélioration en une majoration de E λ . Nos résultats impliquent que le noyau de covariance redimensionné d’une onde aléatoire monochromatique converge localement dans la topologie C ∞ vers une limite d’échelle universelle à un taux logarithmique inverse.

In this paper, we study the two-point Weyl Law for the Laplace–Beltrami operator on a smooth, compact Riemannian manifold M with no conjugate points. That is, we find the asymptotic behavior of the Schwartz kernel, E λ (x,y), of the projection operator from L 2 (M) onto the direct sum of eigenspaces with eigenvalue smaller than λ 2 as λ→∞. In the regime where x,y are restricted to a compact neighborhood of the diagonal in M×M, we obtain a uniform logarithmic improvement in the remainder of the asymptotic expansion for E λ and its derivatives of all orders, which generalizes a result of Bérard, who treated the on-diagonal case E λ (x,x). When x,y avoid a compact neighborhood of the diagonal, we obtain this same improvement in an upper bound for E λ . Our results imply that the rescaled covariance kernel of a monochromatic random wave locally converges in the C ∞ -topology to a universal scaling limit at an inverse logarithmic rate.

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DOI : 10.5802/aif.3598
Classification : 35P20
Keywords: Weyl Laws, Spectral Theory, Riemannian Geometry, Random Waves.
Mot clés : Loi de Weyl, théorie spectrale, géométrie riemannienne, ondes aléatoires
Keeler, Blake 1

1 McGill University, 805 Sherbrooke St. W., Montreal QC (Canada) 
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Keeler, Blake. A logarithmic improvement in the two-point Weyl Law for manifolds without conjugate points. Annales de l'Institut Fourier, Online first, 44 p.

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