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

Blake Keeler

Published 2019-05-13Version 1

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, $E_\lambda(x,y)$, of the projection operator from $L^2(M)$ onto the direct sum of eigenspaces with eigenvalue smaller than $\lambda^2$ as $\lambda \to\infty$. We obtain a uniform logarithmic improvement in the remainder of this asymptotic expansion when the points $x,y$ are close together. This result is a generalization of a work by B\'erard, which treated the on-diagonal case, $E_\lambda(x,x)$. The results in this paper allow us to conclude that the rescaled covariance kernel of a monochromatic random wave on a manifold without conjugate points locally converges to a universal scaling limit at an inverse logarithmic rate.