A logarithmic improvement in the two-point Weyl Law for manifolds without conjugate points

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