In this paper, we study the two-point Weyl Law for the Laplace–Beltrami operator on a smooth, compact Riemannian manifold with no conjugate points. That is, we find the asymptotic behavior of the Schwartz kernel, , of the projection operator from onto the direct sum of eigenspaces with eigenvalue smaller than as . In the regime where are restricted to a compact neighborhood of the diagonal in , we obtain a uniform logarithmic improvement in the remainder of the asymptotic expansion for and its derivatives of all orders, which generalizes a result of Bérard, who treated the on-diagonal case . When avoid a compact neighborhood of the diagonal, we obtain this same improvement in an upper bound for . Our results imply that the rescaled covariance kernel of a monochromatic random wave locally converges in the -topology to a universal scaling limit at an inverse logarithmic rate.
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 sans points conjugués. Cela veut dire que nous donnons le comportement asymptotique du noyau de Schwartz, , de l’opérateur de projection de sur la somme directe des espaces propres pour des valeurs propres inférieures à quand . Dans le régime où sont restreints à un voisinage compact de la diagonale en , on obtient une amélioration logarithmique uniforme dans le reste du développement asymptotique pour et ses dérivées de tous ordres , ce qui généralise un résultat de Bérard, qui a traité le cas diagonal . Lorsque évitent un voisinage compact de la diagonale, on obtient cette même amélioration en une majoration de . Nos résultats impliquent que le noyau de covariance redimensionné d’une onde aléatoire monochromatique converge localement dans la topologie vers une limite d’échelle universelle à un taux logarithmique inverse.
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Keywords: Weyl Laws, Spectral Theory, Riemannian Geometry, Random Waves.
Mot clés : Loi de Weyl, théorie spectrale, géométrie riemannienne, ondes aléatoires
Keeler, Blake 1
@article{AIF_2024__74_2_719_0, author = {Keeler, Blake}, title = {A logarithmic improvement in the two-point {Weyl} {Law} for manifolds without conjugate points}, journal = {Annales de l'Institut Fourier}, pages = {719--762}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {74}, number = {2}, year = {2024}, doi = {10.5802/aif.3598}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3598/} }
TY - JOUR AU - Keeler, Blake TI - A logarithmic improvement in the two-point Weyl Law for manifolds without conjugate points JO - Annales de l'Institut Fourier PY - 2024 SP - 719 EP - 762 VL - 74 IS - 2 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.3598/ DO - 10.5802/aif.3598 LA - en ID - AIF_2024__74_2_719_0 ER -
%0 Journal Article %A Keeler, Blake %T A logarithmic improvement in the two-point Weyl Law for manifolds without conjugate points %J Annales de l'Institut Fourier %D 2024 %P 719-762 %V 74 %N 2 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.3598/ %R 10.5802/aif.3598 %G en %F AIF_2024__74_2_719_0
Keeler, Blake. A logarithmic improvement in the two-point Weyl Law for manifolds without conjugate points. Annales de l'Institut Fourier, Volume 74 (2024) no. 2, pp. 719-762. doi : 10.5802/aif.3598. https://aif.centre-mersenne.org/articles/10.5802/aif.3598/
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