## Exponential sums and nonlinear Schrödinger equations.(English)Zbl 0787.35096

A global result is given for the Cauchy problem, in space dimension $$n=4$$, of the Schrödinger equation $i \partial_ tu+\Delta u+uF(| u |^ 2)=0,\;u \text{ periodic in } x,\;u(x,0)=\varphi(x),$ $$\varphi \in H^ 2(\mathbb{R}^ 4/ \mathbb{Z}^ 4)$$, $$\| \varphi \|_ 2$$ small, $$F(z)\leq cz^{1/2}$$, $$| F'(z)| \leq Cz^{-1/2}$$, $$| F''(z)| \leq Cz^{-3/2}$$.
The global wellposedness is derived from the local wellposedness and the conservation laws. The local result is based on Picard’s fixed point method, using the associated integral equation.