We consider a spectral Galerkin method with a tensor-product Hermite
basis for the linear, multi-dimensional Schrödinger equation with a time-dependent potential. For the resulting
ODE for the expansion coefficients, we propose a fast algorithm to compute directly the action of the stiffness matrix on a vector without actually assembling the matrix itself, as required
in each time step. Together with the application of a hyperbolically reduced basis, this reduces the
computational effort considerably. The
analysis is based on a representation of the three-term recurrence relation for the one-dimensional
basis functions as a full binary tree.