ON INTEGRALS OF EIGENFUNCTIONS OVER GEODESICS

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 R tend to zero in the sense of distributions as the eigenvalue lambda(j) -> , and so integrals of eigen-functions over periodic geodesics tend to zero as lambda(j) -> infinity. The assumption of negative curvature is necessary for the latter result.