LAUSR

Abstract We show that, on compact Riemannian surfaces of nonpositive curvature, the generalized periods, i.e. the ????-th order Fourier coefficients of eigenfunctions eλe_{\lambda} over a closed smooth curve ???? which satisfies a natural curvature condition, go to 0 at the rate of O⁢((log⁡λ)-12)O((\log\lambda)^{-\frac{1}{2}}) in the high energy limit λ→∞\lambda\to\infty if 0<|ν|λ<1-δ0<\frac{\lvert\nu\rvert}{\lambda}<1-\delta for any fixed 0<δ<10<\delta<1. Our result implies, for instance, that the generalized periods over geodesic circles on any surfaces with nonpositive curvature would converge to zero at the rate of O⁢((log⁡λ)-12)O((\log\lambda)^{-\frac{1}{2}}).