Improved generalized periods estimates over curves on Riemannian surfaces with nonpositive curvature
Keyword(s):
Abstract We show that, on compact Riemannian surfaces of nonpositive curvature, the generalized periods, i.e. the 𝜈-th order Fourier coefficients of eigenfunctions e λ e_{\lambda} over a closed smooth curve 𝛾 which satisfies a natural curvature condition, go to 0 at the rate of O ( ( log λ ) - 1 2 ) O((\log\lambda)^{-\frac{1}{2}}) in the high energy limit λ → ∞ \lambda\to\infty if 0 < | ν | λ < 1 - δ 0<\frac{\lvert\nu\rvert}{\lambda}<1-\delta for any fixed 0 < δ < 1 0<\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 ) - 1 2 ) O((\log\lambda)^{-\frac{1}{2}}) .
Keyword(s):
Keyword(s):
1973 ◽
Vol 6
(2)
◽
pp. 236-246
2013 ◽
Vol 04
(09)
◽
pp. 1171-1175
◽
1939 ◽
Vol 170
(941)
◽
pp. 190-205
◽
2013 ◽
Vol 21
◽
pp. 153-154
Keyword(s):
2019 ◽
Vol 34
(27)
◽
pp. 1950152