  In this work, we study the nonlinear spatially homogeneous Lan-dau equation
with Maxwellian molecules, by using the spectral analysis, we show that the non
linear Landau operators is almost linear, and we prove the existence of weak
solution for the Cauchy problem with the initial datum belonging to Shubin
space of negative index which conatins the probability measures. Based on this
spectral decomposition, we prove also that the Cauchy problem enjoys
Gelfand-Shilov smoothing effect, meaning that the weak solution of the Cauchy
problem with Shubin class initial datum is ultra-analytics and exponential
decay for any positive time.
