  In this work, we study the Cauchy problem for the spatially homogeneous
non-cutoff Boltzamnn equation with Maxwellian molecules. We prove that this
Cauchy problem enjoys Gelfand-Shilov regularizing effect, that means the
smoothing properties is same as the Cauchy problem defined by the evolution
equation associated to a fractional harmonic oscillator. The power of this
fractional is exactly the singular index of non-cutoff collisional kernel of
Boltzmann operator. So that we get the regularity of solution in the Gevery
class with the sharp power and the optimal exponential decay of solutions. We
also give a method to construct the solution of the nonlinear Boltzmann
equation by solving an infinite ``triangular'' systems of ordinary differential
equations.The key tools is the spectral decomposition of linear and non-linear
Boltzmann operators.
