  We consider the linear time-dependent Schr\"odinger equation with a
time-dependent smooth potential on an unbounded domain. A Galerkin spectral
method with a tensor-product Hermite basis is used as a discretization in
space. Discretizing the resulting ODE for the Hermite expansion coefficients
involves the computation of the action of the Galerkin matrix on a vector in
each time step. We propose a fast algorithm for the direct computation of this
matrix-vector product without actually assembling the matrix itself. The costs
scale linearly in the size of the basis. Together with the application of a
hyperbolically reduced basis, this reduces the computational effort
considerably and helps cope with the infamous curse of dimensionality. The
application of the fast algorithm is limited to the case of the potential being
significantly smoother than the solution. The error analysis is based on a
binary tree representation of the three-term recurrence relation for the
one-dimensional Hermite functions. The fast algorithm constitutes an efficient
tool for schemes involving the action of a matrix due to spectral
discretization on a vector, and it is also applicable in the context of
spectral approximations for linear problems other than the Schr\"odinger
equation.
