On integrals of eigenfunctions over geodesics

Abstract

If $(M,g)$ is a compact Riemannian surface then the integrals of $L^2(M)$-normalized eigenfunctions $e_j$ over geodesic segments of fixed length are uniformly bounded. Also, if $(M,g)$ has negative curvature and $\gamma (t)$ is a geodesic parameterized by arc length, the measures $e_j(\gamma (t))dt$ on $\mathbb{R}$ tend to zero in the sense of distributions as the eigenvalue $\la_j\to \infty$, and so integrals of eigenfunctions over periodic geodesics tend to zero as $\la_j\to \infty$. The assumption of negative curvature is necessary for the latter result.

10 pages. Final version. To appear in Proceedings of the American Math. Soc