Philippe Charron

Triple products of eigenfunctions

Let (M,g) be a compact manifold. We know that two distinct Laplace-Beltrami eigenfunctions are orthogonal in L^2. Therefore, the Fourier expansion of an eigenfunction (in terms of all eigenfunctions) only has one non-zero term. On the interval, the product of two sine functions is a sum of sines with frequency added and subtracted. Now, what happens for the product of eigenfunctions on a general manifold? In an upcoming work with François Pagano, we show that if the metric g is analytic, something similar happens: any product \phi_j \phi_k can be exponentially well approximated by eigenfunctions with eigenvalues under the sum of the frequencies time some constant.