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}})
.