We show that, on compact Riemannian surfaces of nonpositive curvature, the generalized periods, i.e. the 𝜈-th order Fourier coefficients of eigenfunctions eλe_{\lambda} over a closed smooth curve 𝛾 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}}).

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改进的具有非正曲率的黎曼曲面上的曲线上的广义周期估计

我们表明，在非正曲率的紧致黎曼曲面上，广义周期，即在满足自然曲率条件的闭合光滑曲线over上，本征函数eλe_ {\ lambda}的order阶傅立叶系数以0的速率变为0高能量极限λ→∞\ lambda \ to \ infty中的O⁢（（log⁡λ）-12）O（（\ log \ lambda）^ {-\ frac {1} {2}}）的值如果0 < |ν|λ<1-δ0<\ frac {\ lvert \ nu \ rvert} {\ lambda} <1- \ delta对于任何固定的0 <δ<10 <\ delta <1。例如，我们的结果表明，在任何具有非正曲率的曲面上的测地线圆上的广义​​周期将以O⁢（（log⁡λ）-12）O（（\ log \ lambda）^ {- \ frac {1} {2}}）。