On the concentration of the Fourier coefficients for products of Laplace-Beltrami eigenfunctions on real-analytic manifolds

On a closed analytic manifold $(M,g)$, let φᵢ be the eigenfunctions of Δg with eigenvalues λᵢ² and let f:=∏ φkⱼ be a finite product of Laplace-Beltrami eigenfunctions. We show that f, φᵢ L²(M) decays exponentially as soon as λᵢ > C ∑ λkⱼ for some constant C depending only on M. Moreover, by using a lower bound on \| f \|L²(M), we show that $99%$ of the L²-mass of f can be recovered using only finitely many Fourier coefficients.