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For $n\geq 3$, we study the Cauchy problem for the fourth order nonlinearSchr\"{o}dinger equations, for which the existence of the scattering operatorsand the global well-posedness of solutions with small data in Besov spaces$B^{s}_{2,1}(\R^n)$ are obtained. In one spatial dimension, we get the globalwell-posedness result with small data in the critical homogeneous Besov spaces$\dot{B}^{s}_{2,1}$. As a by-product, the existence of the scattering operatorswith small data is also obtained. In order to show these results, the globalversion of the estimates for the maximal functions and the local smoothingeffects on the fourth order Schr\"{o}dinger semi-groups are established.

Global well-posedness and scattering for the fourth order nonlinear Schr\"{o}dinger equations with small data