In the below reference, we considered Galerkin methods for the high-dimensional Schrödinger equation. We presented a fast algorithm to compute the product of the Galerkin potential matrix times a vector and gave a convergence analysis. Neither does the fast algorithm require assembly of the matrix nor do we need to employ quadrature. This talk presents some generalizations and further applications of the underlying ideas.
Reference: B. Brumm, A fast matrix-free algorithm for spectral approximations to the Schrödinger equation, November 2013, update August 2014.