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 $\phi_i$ be the eigenfunctions of $\Delta_g$ with eigenvalues $\lambda_i^2$ and let $f:=\prod \phi_{k_j}$ be a finite product of Laplace-Beltrami eigenfunctions. We show that $\left\langle f, \phi_i \right\rangle_{L^2(M)}$ decays exponentially as soon as $\lambda_i > C \sum \lambda_{k_j}$ for some constant $C$ depending only on $M$. Moreover, by using a lower bound on $\| f \|_{L^2(M)} $, we show that $99\%$ of the $L^2$-mass of $f$ can be recovered using only finitely many Fourier coefficients.