AbstractWe obtain approximation bounds for products of quasimodes for the Laplace–Beltrami operator on compact Riemannian manifolds of all dimensions without boundary. We approximate the products of quasimodes uv by a low-degree vector space {B_{n}}, and we prove that the size of the space {\dim(B_{n})} is small. In this paper, we first study bilinear quasimode estimates of all dimensions {d=2,3}, {d=4,5} and {d\geq 6}, respectively, to make the highest frequency disappear from the right-hand side. Furthermore, the result of the case {\lambda=\mu} of bilinear quasimode estimates improves {L^{4}} quasimodes estimates of Sogge and Zelditch in [C. D. Sogge and S. Zelditch,
A note on L^{p}-norms of quasi-modes,
Some Topics in Harmonic Analysis and Applications,
Adv. Lect. Math. (ALM) 34,
International Press, Somerville 2016, 385–397] when {d\geq 8}. And on this basis, we give approximation bounds in {H^{-1}}-norm. We also prove approximation bounds for the products of quasimodes in {L^{2}}-norm using the results of {L^{p}}-estimates for quasimodes in [M. Blair, Y. Sire and C. D. Sogge,
Quasimode, eigenfunction and spectral projection bounds for Schrodinger operators on manifolds with critically singular potentials,
preprint 2019, https://arxiv.org/abs/1904.09665]. We extend the results of Lu and Steinerberger in [J. F. Lu and S. Steinerberger,
On pointwise products of elliptic eigenfunctions,
preprint 2018, https://arxiv.org/abs/1810.01024v2] to quasimodes.
Carleman estimates with sharp weights and boundary observability for wave operators with critically singular potentials
