Eigenfunctions restricted to submanifolds and their Fourier coefficients

Abstract: Consider a Laplace-Beltrami eigenfunction on some compact manifold, and restrict it to a compact submanifold. We may write the restricted eigenfunction as a combination of eigenbasis elements intrinsic to the submanifold, whose coefficients we will call Fourier coefficients. What does the spectral decomposition of the restricted eigenfunction look like? How much of the mass of the Fourier coefficients is concentrated near the eigenvalue? Do the Fourier coefficients "feel" the geometry of the submanifold or ambient manifold? If so, how?

I will present joint work with Yakun Xi and Steve Zelditch on such questions. Indeed, various aspects of these Fourier coefficients reflect the geometry of the submanifold and ambient space. Of particular importance are configurations of "geodesic bi-angles," which consist of a pair of geodesics, one in the ambient manifold and one intrinsic to the submanifold, with shared endpoints. These bi-angles arise in the wavefront set analysis a la the Duistermaat-Guillemin theorem.

analysis of PDEsnumber theoryrepresentation theory

Audience: researchers in the topic

Harmonic Analysis and Symmetric Spaces 2021