Fourier coefficients of restrictions of eigenfunctions

Abstract

Let {e_j} be an orthonormal basis of Laplace eigenfunctions of a compact Riemannian manifold (M, g). Let H ⊂ M be a submanifold and {ψ_k} be an orthonormal basis of Laplace eigenfunctions of H with the induced metric. We obtain joint asymptotics for the Fourier coefficients

$${\left\langle {{\gamma _H}{e_j},{\psi _k}} \right\rangle _{{L^2}(H)}} = {\int_H {{e_j}\overline \psi } _k}d{V_H}$$

of restrictions γh^ej of e_j to H. In particular, we obtain asymptotics for the sums of the norm-squares of the Fourier coefficients over the joint spectrum $\left\{ {({\mu _k},{\lambda _j})} \right\}_{j,k - 0}^\infty$ of the (square roots of the) Laplacian Δ_M on M and the Laplacian Δ_H on H in a family of suitably ‘thick’ regions in ℝ². Thick regions include (1) the truncated cone μ_k/λ_j ∈ [a, b] ⊂ (0, 1) and λ_j ≼ λ, and (2) the slowly thickening strip ∣μ_k − cλ_j∣ ≼ w(λ) and λ_j ⩼ λ, where w(λ) is monotonic and 1 ≪ w(λ) ≾ λ^1/2. Key tools for obtaining the asymptotics include the composition calculus of Fourier integral operators and a new multidimensional Tauberian theorem.