On integrals of eigenfunctions over geodesics

2013/02/22 by Xuehua Chen, Christopher D. Sogge, Chen, Xuehua +1 · 3 citations
Mathematics · #Analysis of PDEs (math.AP) #Analytic and geometric function theory #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.1302.5597

openalex publication_date 2013/02/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

If (M,g) is a compact Riemannian surface then the integrals of L²(M)-normalized eigenfunctions e_j over geodesic segments of fixed length are uniformly bounded. Also, if (M,g) has negative curvature and γ(t) is a geodesic parameterized by arc length, the measures e_j(γ(t)) dt on \R tend to zero in the sense of distributions as the eigenvalue \la_j→ ∞, and so integrals of eigenfunctions over periodic geodesics tend to zero as \la_j→ ∞. The assumption of negative curvature is necessary for the latter result.

Cited by

• Kuznecov remainders and generic metrics
• Average estimate for Eigenfunctions along geodesics in the quantum completely integrable case
• Integrals of eigenfunctions over curves in surfaces of nonpositive curvature

Related

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• Integrals of eigenfunctions over curves in surfaces of nonpositive curvature
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• A few endpoint geodesic restriction estimates for eigenfunctions