In this work, we prove the Gelfand-Shilov smoothing effect of the Cauchy problem for the spatially homogeneous non-cutoff Boltzamnn equation with Maxwellian molecules. The smoothing properties is same as the Cauchy problem defined by the evolu-tion equation associated to a fractional harmonic oscillator. We also construct the solution of the Boltzmann equation by solving an infinite systems of ordinary differential equations. The key tools is the spectral decomposition of linear and non-linear Boltzmann operators,