  In this article, we develop and analyse a new spectral method to solve the
semi-classical Schr\"odinger equation based on the Gaussian wave-packet
transform (GWPT) and Hagedorn's semi-classical wave-packets (HWP). The GWPT
equivalently recasts the highly oscillatory wave equation as a much less
oscillatory one (the $w$ equation) coupled with a set of ordinary differential
equations governing the dynamics of the so-called GWPT parameters. The
Hamiltonian of the $ w $ equation consists of a quadratic part and a small
non-quadratic perturbation, which is of order $ \mathcal{O}(\sqrt{\varepsilon
  }) $, where $ \varepsilon\ll 1 $ is the rescaled Planck's constant. By
expanding the solution of the $ w $ equation as a superposition of Hagedorn's
wave-packets, we construct a spectral method while the $
\mathcal{O}(\sqrt{\varepsilon}) $ perturbation part is treated by the Galerkin
approximation. This numerical implementation of the GWPT avoids imposing
artificial boundary conditions and facilitates rigorous numerical analysis. For
arbitrary dimensional cases, we establish how the error of solving the
semi-classical Schr\"odinger equation with the GWPT is determined by the errors
of solving the $ w $ equation and the GWPT parameters. We prove that this
scheme has the spectral convergence with respect to the number of Hagedorn's
wave-packets in one dimension. Extensive numerical tests are provided to
demonstrate the properties of the proposed method.
