Improved Generalized Periods Estimates Over Curves on Riemannian Surfaces with Nonpositive Curvature
Abstract
We show that on compact Riemann surfaces of nonpositive curvature, the generalized periods, i.e. the $\nu$th order Fourier coefficients of eigenfunctions $e_\lambda$ over a closed smooth curve $\gamma$ which satisfies a natural curvature condition, go to 0 at the rate of $O((\log\lambda)^{1/2})$, if $0<\nu/\lambda<1\delta$, for any fixed $0<\delta<1$. Our result implies, for instance, the generalized periods over geodesic circles on any surfaces with nonpositive curvature would converge to zero at the rate of $O((\log\lambda)^{1/2})$. A direct corollary of our results and the QER theorem of Toth and Zelditch is that for a geodesic circle $\gamma$ on a compact hyperbolic surface, the restriction $e_{\lambda_j}_\gamma$ of an orthonormal basis $\{e_{\lambda_j}\}$ has a full density subsequence that goes to zero in weak$L^2(\gamma)$. One key step of our proof is a microlocal decomposition of the measure over $\gamma$ into tangential and transversal parts.
 Publication:

arXiv eprints
 Pub Date:
 June 2018
 arXiv:
 arXiv:1807.00041
 Bibcode:
 2018arXiv180700041W
 Keywords:

 Mathematics  Analysis of PDEs;
 Mathematics  Differential Geometry;
 Mathematics  Number Theory;
 35P20;
 58J51;
 42B37
 EPrint:
 26 pages, 3 figures, 2 corollaries on weak $L^2$ convergence added. These results should be compared with arXiv:1711.09864 by the second author on the closed geodesic case