2019/05/13 by Blake Keeler, Keeler, Blake · 1 citation
Mathematics · #35P20 #Analysis of PDEs (math.AP) #FOS: Mathematics #Random Matrices and Applications #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.1905.05136
openalex publication_date 2019/05/13 · openalex created_date 2022/07/29 · openalex updated_date 2026/07/28
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, Eλ(x,y), of the projection operator from L2(M) onto the direct sum of eigenspaces with eigenvalue smaller than λ2 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 Eλ and its derivatives of all orders, which generalizes a result of Bérard, who treated the on-diagonal case Eλ(x,x). When x,y avoid a compact neighborhood of the diagonal, we obtain this same improvement in an upper bound for Eλ. 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.