缅北强奸

Title: The Two-Point Weyl Law on Manifolds without Conjugate Points

Abstract: In this talk, we discuss the asymptotic behavior of the spectral function of the Laplace-Beltrami operator on a compact Riemannian manifold $M$ with no conjugate points. The spectral function, denoted $\Pi_\lambda(x,y),$ is defined as the Schwartz kernel of the orthogonal projection from $L^2(M)$ onto the eigenspaces with eigenvalue at most $\lambda^2$. In the regime where $(x,y)$ is restricted to a sufficiently small neighborhood of the diagonal in $M\times M$, we obtain a uniform logarithmic improvement in the remainder of the asymptotic expansion for $\Pi_\lambda$ and its derivatives of all orders. This generalizes a result of B\'erard which established an on-diagonal estimate for $\Pi_\lambda(x,x)$ without derivatives. Furthermore, when $(x,y)$ avoids a compact neighborhood of the diagonal, we obtain the same logarithmic improvement in the standard upper bound for the derivatives of $\Pi_\lambda$ itself. We also discuss an application of these results to the study of monochromatic random waves.

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Meeting ID: 817 0613 6500

Passcode: 872949
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